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<b>set </b><font color="Maroon" title="c3">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>;<br/>
<a NAME="E1:1_1_1"/><i><font color="Green" title="E2">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) &amp; ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> ) &amp; ( $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; $2 is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> implies for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/>( (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 implies ( ( <font color="Olive" title="b1">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) &amp; ( (  not <font color="Olive" title="b1">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) &amp; (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> $1 implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) ) &amp; ( $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; $2 is  not   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> implies $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) );<br/>
<a NAME="E2:1_1_1"/><i><font color="Green" title="E3">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1"/>
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c5">x</font> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) or ( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c5">x</font> is  not   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_1"/><i><font color="Green" title="E4">A4</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>take </b>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_2"/><i><font color="Green" title="E4">A5</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>take </b>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_2"><i><font color="Green" title="E4">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_1_1_3"/><i><font color="Green" title="E4">A6</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3"/><i><font color="Green" title="E5">A7</font></i>: 
<font color="Maroon" title="c5">x</font> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_1_1_3"><i><font color="Green" title="E5">A7</font></i></a></i>;<br/><b>thus </b><a NAME="E4:1_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E6">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_1_1_3_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c6">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  not <font color="Maroon" title="c6">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E7">A9</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E8">A10</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E9">A11</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">i1</font>, <font color="Maroon" title="c8">j1</font>, <font color="Maroon" title="c9">i2</font>, <font color="Maroon" title="c10">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E10">A12</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i1</font>,<font color="Maroon" title="c8">j1</font> )
 <b>and </b><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E11">A13</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i2</font>,<font color="Maroon" title="c10">j2</font> )
 <b>and </b><br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E12">A14</font></i>: 
( ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E13">A15</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i1</font> &amp; <font color="Maroon" title="c7">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j1</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E14">A16</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E15">A17</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c8">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E13">A15</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E11:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E16">A18</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E12:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E17">A19</font></i>: 
( <font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E13:1_1_1_1_1_3_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/><a NAME="E14:1_1_1_1_1_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E18">A20</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E19">A21</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a></i>;<br/><b>thus </b><a NAME="E16:1_1_1_1_1_3_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1"/>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E20">A22</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E12">A14</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E21">A23</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E22">A24</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E23">A25</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E24">A26</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E22">A24</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c8">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E21">A23</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,0,<font color="Maroon" title="c8">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E13">A15</font></i></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E21">A23</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E20">A22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E21">A27</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font>;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><i><font color="Green" title="E22">A28</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><i><font color="Green" title="E23">A29</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E23">A29</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E24">A30</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E22">A28</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E21">A27</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E13">A15</font></i></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E21">A27</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green" title="E24">A30</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E20">A22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"/><i><font color="Green" title="E21">A31</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font>;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><i><font color="Green" title="E22">A32</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><i><font color="Green" title="E23">A33</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green" title="E23">A33</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E24">A34</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green" title="E22">A32</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E21">A31</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>,0 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E21">A31</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green" title="E24">A34</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E20">A22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"/><i><font color="Green" title="E21">A35</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><i><font color="Green" title="E22">A36</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><i><font color="Green" title="E23">A37</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green" title="E23">A37</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><b>then </b><i><font color="Green" title="E24">A38</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green" title="E22">A36</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E21">A35</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E21">A35</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green" title="E24">A38</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E20">A22</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E20">A39</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E12">A14</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E21">A40</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font>;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><i><font color="Green" title="E22">A41</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><i><font color="Green" title="E23">A42</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green" title="E23">A42</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E24">A43</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green" title="E22">A41</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E20">A39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E21">A40</font></i></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E15">A17</font></i></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E21">A40</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green" title="E24">A43</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E20">A39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E21">A44</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><i><font color="Green" title="E22">A45</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><i><font color="Green" title="E23">A46</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E23">A46</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E24">A47</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E22">A45</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c8">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E20">A39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E21">A44</font></i></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E13">A15</font></i></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E21">A44</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green" title="E24">A47</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E20">A39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"/><i><font color="Green" title="E21">A48</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><i><font color="Green" title="E22">A49</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><i><font color="Green" title="E23">A50</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green" title="E23">A50</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E24">A51</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green" title="E22">A49</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E20">A39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E21">A48</font></i></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,0,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E15">A17</font></i></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E21">A48</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green" title="E24">A51</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E20">A39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E21">A52</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font>;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><i><font color="Green" title="E22">A53</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><i><font color="Green" title="E23">A54</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green" title="E23">A54</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><b>then </b><i><font color="Green" title="E24">A55</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green" title="E22">A53</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/>
<font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">i1</font>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E20">A39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E21">A52</font></i></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">i1</font>,0 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E13">A15</font></i></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E21">A52</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green" title="E24">A55</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E20">A39</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"/><i><font color="Green" title="E20">A56</font></i>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> ) or ( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E12">A14</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E21">A57</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c10">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><i><font color="Green" title="E22">A58</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><i><font color="Green" title="E23">A59</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E23">A59</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E24">A60</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E22">A58</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E20">A56</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E21">A57</font></i></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="jordan8.html#T19" title="JORDAN8:th.19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E21">A57</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E24">A60</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E20">A56</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E21">A61</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font>;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E22">A62</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E23">A63</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E23">A63</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E24">A64</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E22">A62</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E20">A56</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E21">A61</font></i></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>,0 <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E21">A61</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E24">A64</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E20">A56</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"/><i><font color="Green" title="E21">A65</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font>;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><i><font color="Green" title="E22">A66</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><i><font color="Green" title="E23">A67</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green" title="E23">A67</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E24">A68</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green" title="E22">A66</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/>
<font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/><b>then </b>
( <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E20">A56</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E21">A65</font></i></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E21">A65</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green" title="E24">A68</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E20">A56</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"/><i><font color="Green" title="E21">A69</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>take </b><font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c10">j2</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i1'</font>, <font color="Maroon" title="c15">j1'</font>, <font color="Maroon" title="c16">i2'</font>, <font color="Maroon" title="c17">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><i><font color="Green" title="E22">A70</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><i><font color="Green" title="E23">A71</font></i>: 
( <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1'</font>,<font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green" title="E23">A71</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><b>then </b><i><font color="Green" title="E24">A72</font></i>: 
( <font color="Maroon" title="c7">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i1'</font> &amp; <font color="Maroon" title="c8">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j1'</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c10">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green" title="E22">A70</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/>
<font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54" title="BINARITH:th.54">BINARITH:54</a></i>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E20">A56</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E21">A69</font></i></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,0,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="ref" href="jordan8.html#T21" title="JORDAN8:th.21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/>
( ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c14">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2'</font> &amp; <font color="Maroon" title="c15">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c11">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i2'</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2'</font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E14">A16</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E17">A19</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E21">A69</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green" title="E24">A72</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"/>
<font color="Maroon" title="c11">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"/>
<font color="Maroon" title="c11">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">f1</font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E20">A56</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E7">A73</font></i>: 
(  not <font color="Maroon" title="c6">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E6">A8</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_2"><i><font color="Green" title="E7">A73</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E6">A74</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">k</font>
 ;<br/><div class="add"><b>take </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green" title="E4">A6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E6">A74</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_4"/><i><font color="Green" title="E4">A75</font></i>: 
( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c5">x</font> is  not   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 ;<br/><div class="add"><b>take </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">x</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_4"><i><font color="Green" title="E4">A75</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:1_1_1"/><i><font color="Green" title="E4">A76</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E5:1_1_1"/><i><font color="Green" title="E5">A77</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E6:1_1_1"/><i><font color="Green" title="E6">A78</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="lattice5.html#S1" title="LATTICE5:sch.1">LATTICE5:sch 1</a>(<a class="txt" href="jordan13.html#E2:1_1_1"><i><font color="Green" title="E3">A3</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>
<a NAME="E7:1_1_1"/><i><font color="Green" title="E7">A79</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E8:1_1_1"/><b>then </b><i><font color="Green" title="E8">A80</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green" title="E5">A77</font></i></a></i>;<br/>
<a NAME="E9:1_1_1"/><i><font color="Green" title="E9">A81</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_2"/><i><font color="Green" title="E10">A82</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_2_1"/>
( <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_1"/>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_2_1_1"/>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_2"/>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_2_1_2"/>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_3"/><i><font color="Green" title="E11">A83</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_2_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_2"><i><font color="Green" title="E10">A82</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_2_1_3_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_3_1_1_1"/><i><font color="Green" title="E12">A84</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_2_1_3_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_2_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c6">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  not <font color="Maroon" title="c6">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_3_1_1_1_1_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_2_1_3_1_1_1_1_1_1"/><b>then </b>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3"><i><font color="Green" title="E11">A83</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3_1_1_1"><i><font color="Green" title="E12">A84</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_2_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E13">A85</font></i>: 
(  not <font color="Maroon" title="c6">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c6">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_2_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ;<br/><b>hence </b><a NAME="E3:1_1_1_2_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3"><i><font color="Green" title="E11">A83</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3_1_1_1"><i><font color="Green" title="E12">A84</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3_1_1_1_1_1_2"><i><font color="Green" title="E13">A85</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_3_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_2_1_3_1_1_2"/>
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1"><i><font color="Green" title="E7">A79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3"><i><font color="Green" title="E11">A83</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:1_1_1"/><i><font color="Green" title="E10">A86</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E8:1_1_1"><i><font color="Green" title="E8">A80</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1"><i><font color="Green" title="E9">A81</font></i></a>);<br/></i>
<a NAME="E11:1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_1_3"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font>
 ;<br/>

<a NAME="E2:1_1_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:1_1_1_3"/><b>then </b>
<font color="Maroon" title="c5">y</font> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1"><i><font color="Green" title="E4">A76</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1"><i><font color="Green" title="E10">A86</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:1_1_1_3"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1"><i><font color="Green" title="E4">A76</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<a NAME="E13:1_1_1"/><i><font color="Green" title="E11">A87</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green" title="E5">A77</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E14:1_1_1"/><i><font color="Green" title="E12">A88</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_4"/><i><font color="Green" title="E13">A89</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 ;<br/>

<a NAME="E2:1_1_1_4"/><i><font color="Green" title="E14">A90</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_4_1"/>
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_4_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_4_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_4"><i><font color="Green" title="E14">A90</font></i></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_4_1_2"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_4_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_4"><i><font color="Green" title="E14">A90</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_4_1_3"/><i><font color="Green" title="E15">A91</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_4_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_4_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_4_1_3_1_1"/>
( ( <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  not <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_4_1_3_1_1_1"/>
( <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_4_1_3_1_1_1"/><b>then </b>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_4"><i><font color="Green" title="E13">A89</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_4_1_3"><i><font color="Green" title="E15">A91</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_4_1_3_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_4"><i><font color="Green" title="E13">A89</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_4_1_3_1_1_2"/>
(  not <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> or  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_4_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_4"><i><font color="Green" title="E13">A89</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_4_1_3"><i><font color="Green" title="E15">A91</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_4_1_3_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_4"><i><font color="Green" title="E13">A89</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E15:1_1_1"/><i><font color="Green" title="E13">A92</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E13:1_1_1"><i><font color="Green" title="E11">A87</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1"><i><font color="Green" title="E12">A88</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c6">XS</font> =  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>;<br/>
<b>set </b><font color="Maroon" title="c7">YS</font> =  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>;<br/>
<a NAME="E16:1_1_1"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T58" title="JORDAN1H:th.58">JORDAN1H:58</a></i>;<br/>
<a NAME="E17:1_1_1"/><b>then </b><i><font color="Green" title="E14">A93</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1 <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span> holds <br/>(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) ) );<br/>
<a NAME="E18:1_1_1"/><i><font color="Green" title="E15">A94</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_5"/>
( ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> ) ) &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green" title="E5">A77</font></i></a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>
<b>hence </b><a NAME="E2:1_1_1_5"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0 <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/>

<b>let </b><font color="Maroon" title="c8">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:1_1_1_5"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c8">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> )
 ;<br/>

<b>hence </b><a NAME="E4:1_1_1_5"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span>,<font color="Maroon" title="c8">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span>,<font color="Maroon" title="c8">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green" title="E5">A77</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:1_1_1"/><i><font color="Green" title="E16">A95</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="jordan11.html#T8" title="JORDAN11:th.8">JORDAN11:8</a></i>;<br/>
<a NAME="E20:1_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T58" title="JORDAN1H:th.58">JORDAN1H:58</a></i>;<br/>
<a NAME="E21:1_1_1"/><b>then </b><i><font color="Green" title="E17">A96</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp;  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan1h.html#T58" title="JORDAN1H:th.58">JORDAN1H:58</a>, <a class="ref" href="binarith.html#T73" title="BINARITH:th.73">BINARITH:73</a></i>;<br/>
<a NAME="E22:1_1_1"/><i><font color="Green" title="E18">A97</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_6"/><i><font color="Green" title="E19">A98</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E2:1_1_1_6"/><i><font color="Green" title="E20">A99</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> holds <br/>(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/>

<a NAME="E3:1_1_1_6"/><i><font color="Green" title="E21">A100</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/>
<a NAME="E4:1_1_1_6"/><i><font color="Green" title="E22">A101</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_1"/><i><font color="Green" title="E23">A102</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_1"/><b>then </b><i><font color="Green" title="E24">A103</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a></i>;<br/><div><i><font color="Green" title="E25">A104</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_1_1"/>
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:1_1_1_6_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">l</font> &amp; <font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_1"><i><font color="Green" title="E23">A102</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_1_6_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:1_1_1_6_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_1"><i><font color="Green" title="E24">A103</font></i></a>, <a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E19:1_1_1"><i><font color="Green" title="E16">A95</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_1_2"/>
( <font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E2:1_1_1_6_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">l</font> &amp; <font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_1"><i><font color="Green" title="E23">A102</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_1_6_1_1_2"/><b>then </b>
( <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_1_2"/>
for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">i2</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:1_1_1_6_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6_1_1"><i><font color="Green" title="E25">A104</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>let </b><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E6:1_1_1_6_1_1"/><i><font color="Green" title="E26">A105</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E7:1_1_1_6_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E8:1_1_1_6_1_1"/><b>then </b>
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_1"><i><font color="Green" title="E23">A102</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_1"><i><font color="Green" title="E26">A105</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E9:1_1_1_6_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_6_1_1"><i><font color="Green" title="E26">A105</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_2"/><i><font color="Green" title="E23">A106</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_2"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a></i>;<br/><a NAME="E3:1_1_1_6_1_2"/><b>then </b><i><font color="Green" title="E24">A107</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><a NAME="E4:1_1_1_6_1_2"/><i><font color="Green" title="E25">A108</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="jordan11.html#T8" title="JORDAN11:th.8">JORDAN11:8</a>, <a class="ref" href="jordan11.html#T9" title="JORDAN11:th.9">JORDAN11:9</a></i>;<br/><div><i><font color="Green" title="E26">A109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_2_1"/><i><font color="Green" title="E27">A110</font></i>: 
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:1_1_1_6_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">l</font> &amp; <font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 )
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2"><i><font color="Green" title="E23">A106</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_1_6_1_2_1"/><b>then </b>
( <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_2_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6_1_2"><i><font color="Green" title="E24">A107</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_2"><i><font color="Green" title="E25">A108</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2_1"><i><font color="Green" title="E27">A110</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_2_2"/><i><font color="Green" title="E27">A111</font></i>: 
( <font color="Maroon" title="c9">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E2:1_1_1_6_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">l</font> &amp; <font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 )
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2"><i><font color="Green" title="E23">A106</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_1_6_1_2_2"/><b>then </b><i><font color="Green" title="E28">A112</font></i>: 
( <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></i>;<br/><b>let </b><font color="Maroon" title="c10">i1</font>, <font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">i2</font>, <font color="Maroon" title="c13">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E4:1_1_1_6_1_2_2"/><i><font color="Green" title="E29">A113</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c13">j2</font> )
 ;<br/><a NAME="E5:1_1_1_6_1_2_2"/><i><font color="Green" title="E30">A114</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c12">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <font color="Maroon" title="c13">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2"><i><font color="Green" title="E23">A106</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_2"><i><font color="Green" title="E24">A107</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_2"><i><font color="Green" title="E25">A108</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2_2"><i><font color="Green" title="E27">A111</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_2_2"><i><font color="Green" title="E28">A112</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_2_2"><i><font color="Green" title="E29">A113</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E6:1_1_1_6_1_2_2"/><b>then </b>
<font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i1</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E21:1_1_1"><i><font color="Green" title="E17">A96</font></i></a>, <a class="ref" href="binarith.html#T61" title="BINARITH:th.61">BINARITH:61</a></i>;<br/><a NAME="E7:1_1_1_6_1_2_2"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_2_2"><i><font color="Green" title="E30">A114</font></i></a></i>;<br/><b>hence </b><a NAME="E8:1_1_1_6_1_2_2"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">i2</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_6_1_2"/><i><font color="Green" title="E27">A115</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_2"><i><font color="Green" title="E26">A109</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><b>let </b><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E8:1_1_1_6_1_2"/><i><font color="Green" title="E28">A116</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E9:1_1_1_6_1_2"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:1_1_1_6_1_2"/><b>then </b><i><font color="Green" title="E29">A117</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2"><i><font color="Green" title="E23">A106</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_2"><i><font color="Green" title="E28">A116</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E11:1_1_1_6_1_2"/><i><font color="Green" title="E30">A118</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1"><i><font color="Green" title="E14">A93</font></i></a>, <a class="ref" href="jordan5b.html#T1" title="JORDAN5B:th.1">JORDAN5B:1</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:1_1_1_6_1_2"/><i><font color="Green" title="E31">A119</font></i>: 
 <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E13:1_1_1_6_1_2"/><b>then </b><i><font color="Green" title="E32">A120</font></i>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6_1_2"><i><font color="Green" title="E24">A107</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_2"><i><font color="Green" title="E25">A108</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_2"><i><font color="Green" title="E27">A115</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_2"><i><font color="Green" title="E28">A116</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_6_1_2"><i><font color="Green" title="E29">A117</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_2"><i><font color="Green" title="E30">A118</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><a NAME="E14:1_1_1_6_1_2"/><i><font color="Green" title="E33">A121</font></i>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6_1_2"><i><font color="Green" title="E24">A107</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_2"><i><font color="Green" title="E25">A108</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_2"><i><font color="Green" title="E27">A115</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_2"><i><font color="Green" title="E28">A116</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_6_1_2"><i><font color="Green" title="E29">A117</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_2"><i><font color="Green" title="E30">A118</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_2"><i><font color="Green" title="E31">A119</font></i></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></i>;<br/><b>thus </b><a NAME="E15:1_1_1_6_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_2"><i><font color="Green" title="E32">A120</font></i></a>, <a class="ref" href="jordan11.html#T11" title="JORDAN11:th.11">JORDAN11:11</a></i>;<br/><b>thus </b><a NAME="E16:1_1_1_6_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_2"><i><font color="Green" title="E33">A121</font></i></a>, <a class="ref" href="jordan11.html#T10" title="JORDAN11:th.10">JORDAN11:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3"/><i><font color="Green" title="E23">A122</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_3"/><b>then </b><i><font color="Green" title="E24">A123</font></i>: 
not <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_1_6_1_3"/><i><font color="Green" title="E25">A124</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E4:1_1_1_6_1_3"/><i><font color="Green" title="E26">A125</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">j1</font>, <font color="Maroon" title="c11">i2</font>, <font color="Maroon" title="c12">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:1_1_1_6_1_3"/><i><font color="Green" title="E27">A126</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> )
 <b>and </b><br/><a NAME="E7:1_1_1_6_1_3"/><i><font color="Green" title="E28">A127</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E8:1_1_1_6_1_3"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E9:1_1_1_6_1_3"/><i><font color="Green" title="E29">A128</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j2</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E10:1_1_1_6_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/><a NAME="E11:1_1_1_6_1_3"/><b>then </b><i><font color="Green" title="E30">A129</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:1_1_1_6_1_3"/><i><font color="Green" title="E31">A130</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a></i>;<br/><a NAME="E13:1_1_1_6_1_3"/><i><font color="Green" title="E32">A131</font></i>: 
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6"><i><font color="Green" title="E20">A99</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a></i>;<br/><a NAME="E14:1_1_1_6_1_3"/><b>then </b><i><font color="Green" title="E33">A132</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="ref" href="jordan1h.html#T61" title="JORDAN1H:th.61">JORDAN1H:61</a></i>;<br/><a NAME="E15:1_1_1_6_1_3"/><i><font color="Green" title="E34">A133</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a>, <a class="ref" href="jordan1h.html#T62" title="JORDAN1H:th.62">JORDAN1H:62</a></i>;<br/><a NAME="E16:1_1_1_6_1_3"/><i><font color="Green" title="E35">A134</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a>, <a class="ref" href="jordan1h.html#T63" title="JORDAN1H:th.63">JORDAN1H:63</a></i>;<br/><a NAME="E17:1_1_1_6_1_3"/><i><font color="Green" title="E36">A135</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a>, <a class="ref" href="jordan1h.html#T64" title="JORDAN1H:th.64">JORDAN1H:64</a></i>;<br/><a NAME="E18:1_1_1_6_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ;<br/><a NAME="E19:1_1_1_6_1_3"/><b>then </b><i><font color="Green" title="E37">A136</font></i>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="ref" href="jordan1h.html#T65" title="JORDAN1H:th.65">JORDAN1H:65</a></i>;<br/><a NAME="E20:1_1_1_6_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ;<br/><a NAME="E21:1_1_1_6_1_3"/><b>then </b><i><font color="Green" title="E38">A137</font></i>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="ref" href="jordan1h.html#T66" title="JORDAN1H:th.66">JORDAN1H:66</a></i>;<br/><a NAME="E22:1_1_1_6_1_3"/><i><font color="Green" title="E39">A138</font></i>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="ref" href="jordan1h.html#T67" title="JORDAN1H:th.67">JORDAN1H:67</a></i>;<br/><a NAME="E23:1_1_1_6_1_3"/><i><font color="Green" title="E40">A139</font></i>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="ref" href="jordan1h.html#T68" title="JORDAN1H:th.68">JORDAN1H:68</a></i>;<br/><a NAME="E24:1_1_1_6_1_3"/><i><font color="Green" title="E41">A140</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="ref" href="jordan1h.html#T69" title="JORDAN1H:th.69">JORDAN1H:69</a></i>;<br/><a NAME="E25:1_1_1_6_1_3"/><i><font color="Green" title="E42">A141</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="ref" href="jordan1h.html#T70" title="JORDAN1H:th.70">JORDAN1H:70</a></i>;<br/><a NAME="E26:1_1_1_6_1_3"/><i><font color="Green" title="E43">A142</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="ref" href="jordan1h.html#T71" title="JORDAN1H:th.71">JORDAN1H:71</a></i>;<br/><a NAME="E27:1_1_1_6_1_3"/><i><font color="Green" title="E44">A143</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="ref" href="jordan1h.html#T72" title="JORDAN1H:th.72">JORDAN1H:72</a></i>;<br/><b>thus </b><a NAME="E28:1_1_1_6_1_3"/><i><font color="Green" title="E45">A144</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1_3_1_1"/>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_1_1_1"/>
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c14">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_1_1_1"/><i><font color="Green" title="E46">A145</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_1_1_1"/><i><font color="Green" title="E47">A146</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c15">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><a NAME="E5:1_1_1_6_1_3_1_1_1"/><i><font color="Green" title="E48">A147</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3"><i><font color="Green" title="E30">A129</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:1_1_1_6_1_3_1_1_1"/><i><font color="Green" title="E49">A148</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:1_1_1_6_1_3_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green" title="E31">A130</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_1_1_1"><i><font color="Green" title="E46">A145</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_1_1_1"><i><font color="Green" title="E48">A147</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E50">A149</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E51">A150</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E52">A151</font></i>: 
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green" title="E33">A132</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A149</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_6_1_3"><i><font color="Green" title="E29">A128</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_1_3_1_1_1_1_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_1_1"><i><font color="Green" title="E52">A151</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_1_3_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green" title="E33">A132</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_1"><i><font color="Green" title="E47">A146</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_1"><i><font color="Green" title="E49">A148</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_1"><i><font color="Green" title="E50">A149</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_1"><i><font color="Green" title="E51">A150</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E50">A152</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E51">A153</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3"><i><font color="Green" title="E34">A133</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_2"><i><font color="Green" title="E50">A152</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_1_1_1_2_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3"><i><font color="Green" title="E34">A133</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_1"><i><font color="Green" title="E47">A146</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_1"><i><font color="Green" title="E49">A148</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_2"><i><font color="Green" title="E50">A152</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_2"><i><font color="Green" title="E51">A153</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_3"/><i><font color="Green" title="E50">A154</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_3"/><i><font color="Green" title="E51">A155</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E52">A156</font></i>: 
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_6_1_3"><i><font color="Green" title="E35">A134</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_3"><i><font color="Green" title="E50">A154</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c17">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_6_1_3"><i><font color="Green" title="E29">A128</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_1_3_1_1_1_1_1_3_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_3_1"><i><font color="Green" title="E52">A156</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_1_3_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_1_1_1_3"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_6_1_3"><i><font color="Green" title="E35">A134</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_1"><i><font color="Green" title="E47">A146</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_1"><i><font color="Green" title="E49">A148</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_3"><i><font color="Green" title="E50">A154</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_3"><i><font color="Green" title="E51">A155</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_4"/><i><font color="Green" title="E50">A157</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_4"/><i><font color="Green" title="E51">A158</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_4_1"/><b>then </b>
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_6_1_3"><i><font color="Green" title="E36">A135</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_4"><i><font color="Green" title="E50">A157</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_1_1_1_4_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_1_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_1_1_1_4"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_6_1_3"><i><font color="Green" title="E36">A135</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_1"><i><font color="Green" title="E47">A146</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_1"><i><font color="Green" title="E49">A148</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_4"><i><font color="Green" title="E50">A157</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_4"><i><font color="Green" title="E51">A158</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_1_1_2"/><i><font color="Green" title="E46">A159</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c14">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_1_1_2"/><i><font color="Green" title="E47">A160</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_1_1_2"/><i><font color="Green" title="E48">A161</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c15">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><a NAME="E5:1_1_1_6_1_3_1_1_2"/><i><font color="Green" title="E49">A162</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3"><i><font color="Green" title="E30">A129</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:1_1_1_6_1_3_1_1_2"/><i><font color="Green" title="E50">A163</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:1_1_1_6_1_3_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green" title="E31">A130</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E47">A160</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E49">A162</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_1"/><i><font color="Green" title="E51">A164</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_1"/><i><font color="Green" title="E52">A165</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_6_1_3"><i><font color="Green" title="E37">A136</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E46">A159</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_1"><i><font color="Green" title="E51">A164</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_2_1_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_6_1_3"><i><font color="Green" title="E37">A136</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E46">A159</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E48">A161</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E50">A163</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_1"><i><font color="Green" title="E51">A164</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_1"><i><font color="Green" title="E52">A165</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_2"/><i><font color="Green" title="E51">A166</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_2"/><i><font color="Green" title="E52">A167</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E21:1_1_1_6_1_3"><i><font color="Green" title="E38">A137</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E46">A159</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_2"><i><font color="Green" title="E51">A166</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_2_1_1_2_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E21:1_1_1_6_1_3"><i><font color="Green" title="E38">A137</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E46">A159</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E48">A161</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E50">A163</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_2"><i><font color="Green" title="E51">A166</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_2"><i><font color="Green" title="E52">A167</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_3"/><i><font color="Green" title="E51">A168</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_3"/><i><font color="Green" title="E52">A169</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E53">A170</font></i>: 
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_6_1_3"><i><font color="Green" title="E39">A138</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E46">A159</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_3"><i><font color="Green" title="E51">A168</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_2_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c16">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_6_1_3"><i><font color="Green" title="E29">A128</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_1_3_1_1_2_1_1_3_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_3_1"><i><font color="Green" title="E53">A170</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_1_3_1_1_2_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_2_1_1_3"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_6_1_3"><i><font color="Green" title="E39">A138</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E46">A159</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E48">A161</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E50">A163</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_3"><i><font color="Green" title="E51">A168</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_3"><i><font color="Green" title="E52">A169</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_4"/><i><font color="Green" title="E51">A171</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_4"/><i><font color="Green" title="E52">A172</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_4_1"/><b>then </b><i><font color="Green" title="E53">A173</font></i>: 
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_6_1_3"><i><font color="Green" title="E40">A139</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E46">A159</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_4"><i><font color="Green" title="E51">A171</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_2_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c17">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_6_1_3"><i><font color="Green" title="E29">A128</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_1_3_1_1_2_1_1_4_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_4_1"><i><font color="Green" title="E53">A173</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_1_3_1_1_2_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_2_1_1_4"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_6_1_3"><i><font color="Green" title="E40">A139</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E46">A159</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E48">A161</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_2"><i><font color="Green" title="E50">A163</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_4"><i><font color="Green" title="E51">A171</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_4"><i><font color="Green" title="E52">A172</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_1_1_3"/><i><font color="Green" title="E46">A174</font></i>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c14">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_1_1_3"/><i><font color="Green" title="E47">A175</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_1_1_3"/><i><font color="Green" title="E48">A176</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c15">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/><a NAME="E5:1_1_1_6_1_3_1_1_3"/><i><font color="Green" title="E49">A177</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3"><i><font color="Green" title="E30">A129</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:1_1_1_6_1_3_1_1_3"/><i><font color="Green" title="E50">A178</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:1_1_1_6_1_3_1_1_3"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green" title="E31">A130</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E47">A175</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E49">A177</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_1"/><i><font color="Green" title="E51">A179</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_1"/><i><font color="Green" title="E52">A180</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_6_1_3"><i><font color="Green" title="E41">A140</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E46">A174</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_1"><i><font color="Green" title="E51">A179</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_3_1_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_3_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_6_1_3"><i><font color="Green" title="E41">A140</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E46">A174</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E48">A176</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E50">A178</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_1"><i><font color="Green" title="E51">A179</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_1"><i><font color="Green" title="E52">A180</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_2"/><i><font color="Green" title="E51">A181</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_2"/><i><font color="Green" title="E52">A182</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_2_1"/><b>then </b><i><font color="Green" title="E53">A183</font></i>: 
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_6_1_3"><i><font color="Green" title="E42">A141</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E46">A174</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_2"><i><font color="Green" title="E51">A181</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_3_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_6_1_3"><i><font color="Green" title="E29">A128</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_1_3_1_1_3_1_1_2_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_2_1"><i><font color="Green" title="E53">A183</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_1_3_1_1_3_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_3_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_6_1_3"><i><font color="Green" title="E42">A141</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E46">A174</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E48">A176</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E50">A178</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_2"><i><font color="Green" title="E51">A181</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_2"><i><font color="Green" title="E52">A182</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_3"/><i><font color="Green" title="E51">A184</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_3"/><i><font color="Green" title="E52">A185</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_3_1"/><b>then </b>
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_6_1_3"><i><font color="Green" title="E43">A142</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E46">A174</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_3"><i><font color="Green" title="E51">A184</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_3_1_1_3_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_3_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_3_1_1_3"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_6_1_3"><i><font color="Green" title="E43">A142</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E46">A174</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E48">A176</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E50">A178</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_3"><i><font color="Green" title="E51">A184</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_3"><i><font color="Green" title="E52">A185</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_4"/><i><font color="Green" title="E51">A186</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_4"/><i><font color="Green" title="E52">A187</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">i1'</font>, <font color="Maroon" title="c17">j1'</font>, <font color="Maroon" title="c18">i2'</font>, <font color="Maroon" title="c19">j2'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1'</font>,<font color="Maroon" title="c17">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2'</font>,<font color="Maroon" title="c19">j2'</font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_4_1"/><b>then </b><i><font color="Green" title="E53">A188</font></i>: 
( <font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1'</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2'</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2'</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_6_1_3"><i><font color="Green" title="E44">A143</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E46">A174</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_4"><i><font color="Green" title="E51">A186</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_3_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c16">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_6_1_3"><i><font color="Green" title="E29">A128</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_1_3_1_1_3_1_1_4_1"/><b>then </b>
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_4_1"><i><font color="Green" title="E53">A188</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_1_3_1_1_3_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13" title="UNIFORM1:th.13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_3_1_1_4"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3"><i><font color="Green" title="E24">A123</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_6_1_3"><i><font color="Green" title="E44">A143</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E46">A174</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E48">A176</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_3"><i><font color="Green" title="E50">A178</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_4"><i><font color="Green" title="E51">A186</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_4"><i><font color="Green" title="E52">A187</font></i></a>, <a class="ref" href="jordan8.html#T9" title="JORDAN8:th.9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>let </b><font color="Maroon" title="c13">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E29:1_1_1_6_1_3"/><i><font color="Green" title="E46">A189</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1_3_2_1"/>
( <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_1"/><i><font color="Green" title="E47">A190</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_3_2_1_1"/><i><font color="Green" title="E48">A191</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_6_1_3"><i><font color="Green" title="E29">A128</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_2_1_1"/><i><font color="Green" title="E49">A192</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_6_1_3"><i><font color="Green" title="E29">A128</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><b>thus </b><a NAME="E4:1_1_1_6_1_3_2_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1"/>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_1"/><i><font color="Green" title="E50">A193</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c14">i</font>, <font color="Maroon" title="c15">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_1"/><i><font color="Green" title="E51">A194</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_1"/><i><font color="Green" title="E52">A195</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_1"/><i><font color="Green" title="E53">A196</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3"><i><font color="Green" title="E30">A129</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E52">A195</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green" title="E31">A130</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E51">A194</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E52">A195</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E53">A196</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E54">A197</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E55">A198</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E54">A197</font></i></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green" title="E33">A132</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E48">A191</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E50">A193</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E53">A196</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E54">A197</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E55">A198</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E54">A197</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green" title="E33">A132</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E48">A191</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E53">A196</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E54">A197</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green" title="E55">A198</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E54">A199</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E55">A200</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E54">A199</font></i></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3"><i><font color="Green" title="E34">A133</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E50">A193</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E53">A196</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E54">A199</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E55">A200</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E54">A199</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3"><i><font color="Green" title="E34">A133</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E53">A196</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E54">A199</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green" title="E55">A200</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/><i><font color="Green" title="E54">A201</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/><i><font color="Green" title="E55">A202</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E54">A201</font></i></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_6_1_3"><i><font color="Green" title="E35">A134</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E49">A192</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E50">A193</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E53">A196</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E54">A201</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E55">A202</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E54">A201</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_6_1_3"><i><font color="Green" title="E35">A134</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E49">A192</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E53">A196</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E54">A201</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green" title="E55">A202</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/><i><font color="Green" title="E54">A203</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/><i><font color="Green" title="E55">A204</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E54">A203</font></i></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_6_1_3"><i><font color="Green" title="E36">A135</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E50">A193</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E53">A196</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E54">A203</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E55">A204</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E54">A203</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_6_1_3"><i><font color="Green" title="E36">A135</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green" title="E53">A196</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E54">A203</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green" title="E55">A204</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_6_1_3_2_1_1_1_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_2"/><i><font color="Green" title="E50">A205</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c14">i</font>, <font color="Maroon" title="c15">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_2"/><i><font color="Green" title="E51">A206</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_2"/><i><font color="Green" title="E52">A207</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_2"/><i><font color="Green" title="E53">A208</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3"><i><font color="Green" title="E30">A129</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E52">A207</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green" title="E31">A130</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E51">A206</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E52">A207</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E53">A208</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E54">A209</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E55">A210</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c12">j2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E54">A209</font></i></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_6_1_3"><i><font color="Green" title="E37">A136</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E50">A205</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E53">A208</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E54">A209</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E55">A210</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E54">A209</font></i></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_6_1_3"><i><font color="Green" title="E37">A136</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E50">A205</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E53">A208</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E54">A209</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E55">A210</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E54">A211</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E55">A212</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E54">A211</font></i></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E21:1_1_1_6_1_3"><i><font color="Green" title="E38">A137</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E50">A205</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E53">A208</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E54">A211</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E55">A212</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E54">A211</font></i></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E21:1_1_1_6_1_3"><i><font color="Green" title="E38">A137</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E50">A205</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E53">A208</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E54">A211</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E55">A212</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/><i><font color="Green" title="E54">A213</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/><i><font color="Green" title="E55">A214</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E54">A213</font></i></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_6_1_3"><i><font color="Green" title="E39">A138</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E48">A191</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E50">A205</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E53">A208</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E54">A213</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E55">A214</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E54">A213</font></i></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_6_1_3"><i><font color="Green" title="E39">A138</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E48">A191</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E50">A205</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E53">A208</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E54">A213</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green" title="E55">A214</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E54">A215</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E55">A216</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E54">A215</font></i></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_6_1_3"><i><font color="Green" title="E40">A139</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E49">A192</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E50">A205</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E53">A208</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E54">A215</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E55">A216</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E54">A215</font></i></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_6_1_3"><i><font color="Green" title="E40">A139</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E49">A192</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E50">A205</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green" title="E53">A208</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E54">A215</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green" title="E55">A216</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_6_1_3_2_1_1_1_1_2"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_3"/><i><font color="Green" title="E50">A217</font></i>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c14">i</font>, <font color="Maroon" title="c15">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_3"/><i><font color="Green" title="E51">A218</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_3"/><i><font color="Green" title="E52">A219</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_3"/><i><font color="Green" title="E53">A220</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3"><i><font color="Green" title="E30">A129</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E52">A219</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green" title="E31">A130</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E51">A218</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E52">A219</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E53">A220</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E54">A221</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E55">A222</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/>
( <font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E54">A221</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_6_1_3"><i><font color="Green" title="E41">A140</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E50">A217</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E53">A220</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E54">A221</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E55">A222</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E54">A221</font></i></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_6_1_3"><i><font color="Green" title="E41">A140</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E50">A217</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E53">A220</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E54">A221</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green" title="E55">A222</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E54">A223</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E55">A224</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/>
( <font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i1</font> &amp;  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c9">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E54">A223</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_6_1_3"><i><font color="Green" title="E42">A141</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E49">A192</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E50">A217</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E53">A220</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E54">A223</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E55">A224</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E54">A223</font></i></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_6_1_3"><i><font color="Green" title="E42">A141</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E49">A192</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E50">A217</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E53">A220</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E54">A223</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green" title="E55">A224</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/><i><font color="Green" title="E54">A225</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/><i><font color="Green" title="E55">A226</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E54">A225</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_6_1_3"><i><font color="Green" title="E43">A142</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E50">A217</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E53">A220</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E54">A225</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E55">A226</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E54">A225</font></i></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_6_1_3"><i><font color="Green" title="E43">A142</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E50">A217</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E53">A220</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E54">A225</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green" title="E55">A226</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/><i><font color="Green" title="E54">A227</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/><i><font color="Green" title="E55">A228</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E54">A227</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_6_1_3"><i><font color="Green" title="E44">A143</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E48">A191</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E50">A217</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E53">A220</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E54">A227</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E55">A228</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green" title="E25">A124</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green" title="E26">A125</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3"><i><font color="Green" title="E27">A126</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E54">A227</font></i></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green" title="E28">A127</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_6_1_3"><i><font color="Green" title="E44">A143</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E47">A190</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green" title="E48">A191</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E50">A217</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green" title="E53">A220</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E54">A227</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green" title="E55">A228</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_6_1_3_2_1_1_1_1_3"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2"/>
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_3_2_1_2"/><b>then </b>
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_2_1_2"/><b>then </b><i><font color="Green" title="E47">A229</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green" title="E22">A101</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">i1</font>, <font color="Maroon" title="c15">j1</font>, <font color="Maroon" title="c16">i2</font>, <font color="Maroon" title="c17">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:1_1_1_6_1_3_2_1_2"/><i><font color="Green" title="E48">A230</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1</font>,<font color="Maroon" title="c15">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1</font>,<font color="Maroon" title="c15">j1</font> )
 <b>and </b><br/><a NAME="E6:1_1_1_6_1_3_2_1_2"/><i><font color="Green" title="E49">A231</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2</font>,<font color="Maroon" title="c17">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2</font>,<font color="Maroon" title="c17">j2</font> )
 <b>and </b><br/><a NAME="E7:1_1_1_6_1_3_2_1_2"/><i><font color="Green" title="E50">A232</font></i>: 
( ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_6_1_3_2_1_2"/><i><font color="Green" title="E51">A233</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E9:1_1_1_6_1_3_2_1_2"/>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E47">A229</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:1_1_1_6_1_3_2_1_2"/><b>then </b><i><font color="Green" title="E52">A234</font></i>: 
( <font color="Maroon" title="c13">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> &amp; <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E47">A229</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E51">A233</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:1_1_1_6_1_3_2_1_2"/><i><font color="Green" title="E53">A235</font></i>: 
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6"><i><font color="Green" title="E20">A99</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E47">A229</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1_3_2_1_2_1_1"/>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_1_1_1"/>
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c19">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_2_1_1_1"/><i><font color="Green" title="E54">A236</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c20">i</font> = <font color="Maroon" title="c18">i</font>;<br/><b>take </b><font color="Maroon" title="c21">j</font> = <font color="Maroon" title="c19">j</font>;<br/><b>thus </b><a NAME="E5:1_1_1_6_1_3_2_1_2_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_2_1_1_1"><i><font color="Green" title="E54">A236</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_1_1_2"/>
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c19">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_2_1_1_2"/><i><font color="Green" title="E54">A237</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c20">i</font> = <font color="Maroon" title="c18">i</font>;<br/><b>take </b><font color="Maroon" title="c21">j</font> = <font color="Maroon" title="c19">j</font>;<br/><b>thus </b><a NAME="E5:1_1_1_6_1_3_2_1_2_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_2_1_1_2"><i><font color="Green" title="E54">A237</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_1_1_3"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c19">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_2_1_1_3"/><i><font color="Green" title="E54">A238</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green" title="E21">A100</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green" title="E23">A122</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green" title="E32">A131</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c20">i</font> = <font color="Maroon" title="c18">i</font>;<br/><b>take </b><font color="Maroon" title="c21">j</font> = <font color="Maroon" title="c19">j</font>;<br/><b>thus </b><a NAME="E5:1_1_1_6_1_3_2_1_2_1_1_3"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_2_1_1_3"><i><font color="Green" title="E54">A238</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c19">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E14:1_1_1_6_1_3_2_1_2"/><i><font color="Green" title="E54">A239</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c19">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ;<br/><a NAME="E15:1_1_1_6_1_3_2_1_2"/><i><font color="Green" title="E55">A240</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i1</font>,<font color="Maroon" title="c15">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i2</font>,<font color="Maroon" title="c17">j2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E48">A230</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E49">A231</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E52">A234</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E54">A239</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1_3_2_1_2_2_1"/>
( ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E50">A232</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_2_1_1"/><i><font color="Green" title="E56">A241</font></i>: 
( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_3_2_1_2_2_1_1"/><b>then </b>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j1</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c14">i1</font>,<font color="Maroon" title="c15">j1</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E47">A229</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E48">A230</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E49">A231</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_6_1_3_2_1_2_2_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E48">A230</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E49">A231</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E53">A235</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E55">A240</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_2_2_1_1"><i><font color="Green" title="E56">A241</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_2_1_2"/><i><font color="Green" title="E56">A242</font></i>: 
( <font color="Maroon" title="c14">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_3_2_1_2_2_1_2"/><b>then </b>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c14">i1</font>,<font color="Maroon" title="c15">j1</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c14">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E47">A229</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E48">A230</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E49">A231</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_6_1_3_2_1_2_2_1_2"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E48">A230</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E49">A231</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E53">A235</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E55">A240</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_2_2_1_2"><i><font color="Green" title="E56">A242</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_2_1_3"/><i><font color="Green" title="E56">A243</font></i>: 
( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_3_2_1_2_2_1_3"/><b>then </b>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c16">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c16">i2</font>,<font color="Maroon" title="c17">j2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E47">A229</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E48">A230</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E49">A231</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_6_1_3_2_1_2_2_1_3"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E48">A230</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E49">A231</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E53">A235</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E55">A240</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_2_2_1_3"><i><font color="Green" title="E56">A243</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_2_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_2_1_4"/><i><font color="Green" title="E56">A244</font></i>: 
( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_3_2_1_2_2_1_4"/><b>then </b>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c16">i2</font>,<font color="Maroon" title="c17">j2</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green" title="E19">A98</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E47">A229</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E48">A230</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E49">A231</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_6_1_3_2_1_2_2_1_4"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green" title="E45">A144</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green" title="E46">A189</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E48">A230</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E49">A231</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E53">A235</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3_2_1_2"><i><font color="Green" title="E55">A240</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_2_2_1_4"><i><font color="Green" title="E56">A244</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:1_1_1_6_1_3_2_1_2"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E31:1_1_1_6_1_3"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E23:1_1_1"/><i><font color="Green" title="E19">A245</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E18:1_1_1"><i><font color="Green" title="E15">A94</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1"><i><font color="Green" title="E18">A97</font></i></a>);<br/></i>
<a NAME="E24:1_1_1"/><i><font color="Green" title="E20">A246</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0,<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0,<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a></i>;<br/>
<a NAME="E25:1_1_1"/><i><font color="Green" title="E21">A247</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1,<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 1,<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a></i>;<br/>
<a NAME="E26:1_1_1"/><i><font color="Green" title="E22">A248</font></i>: 
for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b5">j2</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">i2</font>, <font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_7"/><i><font color="Green" title="E23">A249</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:1_1_1_7"/><i><font color="Green" title="E24">A250</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font> )
 <b>and </b><br/><a NAME="E3:1_1_1_7"/><i><font color="Green" title="E25">A251</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font> )
 ;<br/>

<a NAME="E4:1_1_1_7"/><i><font color="Green" title="E26">A252</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/>
<a NAME="E5:1_1_1_7"/><i><font color="Green" title="E27">A253</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/>
<a NAME="E6:1_1_1_7"/><i><font color="Green" title="E28">A254</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/>
<a NAME="E7:1_1_1_7"/><b>then </b><i><font color="Green" title="E29">A255</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_7"><i><font color="Green" title="E23">A249</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/>
<a NAME="E8:1_1_1_7"/><i><font color="Green" title="E30">A256</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_7"><i><font color="Green" title="E23">A249</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7"><i><font color="Green" title="E28">A254</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E9:1_1_1_7"/><b>then </b><i><font color="Green" title="E31">A257</font></i>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_7"><i><font color="Green" title="E29">A255</font></i></a></i>;<br/>
<a NAME="E10:1_1_1_7"/><i><font color="Green" title="E32">A258</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_7"><i><font color="Green" title="E23">A249</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7"><i><font color="Green" title="E26">A252</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7"><i><font color="Green" title="E28">A254</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_7"><i><font color="Green" title="E29">A255</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:1_1_1_7"/><i><font color="Green" title="E33">A259</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_7"><i><font color="Green" title="E30">A256</font></i></a></i>;<br/>
<a NAME="E12:1_1_1_7"/><i><font color="Green" title="E34">A260</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">i1</font> &amp; <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">j1</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">j2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:1_1_1_7_1"/>
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c14">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_7_1"/><i><font color="Green" title="E35">A261</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:1_1_1_7_1"/><i><font color="Green" title="E36">A262</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_7"><i><font color="Green" title="E23">A249</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_7"><i><font color="Green" title="E27">A253</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7"><i><font color="Green" title="E28">A254</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_7"><i><font color="Green" title="E31">A257</font></i></a></i>;<br/><a NAME="E5:1_1_1_7_1"/><i><font color="Green" title="E37">A263</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_7"><i><font color="Green" title="E24">A250</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7"><i><font color="Green" title="E25">A251</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_7"><i><font color="Green" title="E32">A258</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green" title="E36">A262</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:1_1_1_7_1"/><i><font color="Green" title="E38">A264</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green" title="E36">A262</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:1_1_1_7_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_7_1"><i><font color="Green" title="E35">A261</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green" title="E36">A262</font></i></a></i>;<br/><b>thus </b><a NAME="E8:1_1_1_7_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_7"><i><font color="Green" title="E24">A250</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7"><i><font color="Green" title="E25">A251</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_7"><i><font color="Green" title="E30">A256</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_7"><i><font color="Green" title="E33">A259</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_7"><i><font color="Green" title="E34">A260</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7_1"><i><font color="Green" title="E35">A261</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green" title="E36">A262</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_7_1"><i><font color="Green" title="E37">A263</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7_1"><i><font color="Green" title="E38">A264</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_7_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_7"><i><font color="Green" title="E24">A250</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7"><i><font color="Green" title="E25">A251</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_7"><i><font color="Green" title="E30">A256</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_7"><i><font color="Green" title="E33">A259</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_7"><i><font color="Green" title="E34">A260</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7_1"><i><font color="Green" title="E35">A261</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green" title="E36">A262</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_7_1"><i><font color="Green" title="E37">A263</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7_1"><i><font color="Green" title="E38">A264</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E10:1_1_1_7_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_7"><i><font color="Green" title="E24">A250</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7"><i><font color="Green" title="E25">A251</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_7"><i><font color="Green" title="E30">A256</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_7"><i><font color="Green" title="E33">A259</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_7"><i><font color="Green" title="E34">A260</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7_1"><i><font color="Green" title="E35">A261</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green" title="E36">A262</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_7_1"><i><font color="Green" title="E37">A263</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7_1"><i><font color="Green" title="E38">A264</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E11:1_1_1_7_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_7"><i><font color="Green" title="E24">A250</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7"><i><font color="Green" title="E25">A251</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_7"><i><font color="Green" title="E30">A256</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_7"><i><font color="Green" title="E33">A259</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_7"><i><font color="Green" title="E34">A260</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7_1"><i><font color="Green" title="E35">A261</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green" title="E36">A262</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_7_1"><i><font color="Green" title="E37">A263</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7_1"><i><font color="Green" title="E38">A264</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E27:1_1_1"/><i><font color="Green" title="E23">A265</font></i>: 
for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b5">j2</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">i2</font>, <font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_8"/><i><font color="Green" title="E24">A266</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:1_1_1_8"/><i><font color="Green" title="E25">A267</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font> )
 <b>and </b><br/><a NAME="E3:1_1_1_8"/><i><font color="Green" title="E26">A268</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font> )
 ;<br/>

<a NAME="E4:1_1_1_8"/><i><font color="Green" title="E27">A269</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/>
<a NAME="E5:1_1_1_8"/><i><font color="Green" title="E28">A270</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/>
<a NAME="E6:1_1_1_8"/><i><font color="Green" title="E29">A271</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/>
<a NAME="E7:1_1_1_8"/><b>then </b><i><font color="Green" title="E30">A272</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_8"><i><font color="Green" title="E24">A266</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/>
<a NAME="E8:1_1_1_8"/><i><font color="Green" title="E31">A273</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_8"><i><font color="Green" title="E24">A266</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8"><i><font color="Green" title="E29">A271</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E9:1_1_1_8"/><b>then </b><i><font color="Green" title="E32">A274</font></i>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_8"><i><font color="Green" title="E30">A272</font></i></a></i>;<br/>
<a NAME="E10:1_1_1_8"/><i><font color="Green" title="E33">A275</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_8"><i><font color="Green" title="E24">A266</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8"><i><font color="Green" title="E27">A269</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8"><i><font color="Green" title="E29">A271</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_8"><i><font color="Green" title="E30">A272</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:1_1_1_8"/><i><font color="Green" title="E34">A276</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_8"><i><font color="Green" title="E31">A273</font></i></a></i>;<br/>
<a NAME="E12:1_1_1_8"/><i><font color="Green" title="E35">A277</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">i1</font> &amp; <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">j1</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">j2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:1_1_1_8_1"/>
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c14">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_8_1"/><i><font color="Green" title="E36">A278</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E4:1_1_1_8_1"/><i><font color="Green" title="E37">A279</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_8"><i><font color="Green" title="E24">A266</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_8"><i><font color="Green" title="E28">A270</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8"><i><font color="Green" title="E29">A271</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_8"><i><font color="Green" title="E32">A274</font></i></a></i>;<br/><a NAME="E5:1_1_1_8_1"/><i><font color="Green" title="E38">A280</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_8"><i><font color="Green" title="E25">A267</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8"><i><font color="Green" title="E26">A268</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_8"><i><font color="Green" title="E33">A275</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green" title="E37">A279</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:1_1_1_8_1"/><i><font color="Green" title="E39">A281</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green" title="E37">A279</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:1_1_1_8_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_8_1"><i><font color="Green" title="E36">A278</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green" title="E37">A279</font></i></a></i>;<br/><b>thus </b><a NAME="E8:1_1_1_8_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_8"><i><font color="Green" title="E25">A267</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8"><i><font color="Green" title="E26">A268</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_8"><i><font color="Green" title="E31">A273</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_8"><i><font color="Green" title="E34">A276</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_8"><i><font color="Green" title="E35">A277</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8_1"><i><font color="Green" title="E36">A278</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green" title="E37">A279</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_8_1"><i><font color="Green" title="E38">A280</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8_1"><i><font color="Green" title="E39">A281</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_8_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_8"><i><font color="Green" title="E25">A267</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8"><i><font color="Green" title="E26">A268</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_8"><i><font color="Green" title="E31">A273</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_8"><i><font color="Green" title="E34">A276</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_8"><i><font color="Green" title="E35">A277</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8_1"><i><font color="Green" title="E36">A278</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green" title="E37">A279</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_8_1"><i><font color="Green" title="E38">A280</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8_1"><i><font color="Green" title="E39">A281</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E10:1_1_1_8_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_8"><i><font color="Green" title="E25">A267</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8"><i><font color="Green" title="E26">A268</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_8"><i><font color="Green" title="E31">A273</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_8"><i><font color="Green" title="E34">A276</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_8"><i><font color="Green" title="E35">A277</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8_1"><i><font color="Green" title="E36">A278</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green" title="E37">A279</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_8_1"><i><font color="Green" title="E38">A280</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8_1"><i><font color="Green" title="E39">A281</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E11:1_1_1_8_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_8"><i><font color="Green" title="E25">A267</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8"><i><font color="Green" title="E26">A268</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_8"><i><font color="Green" title="E31">A273</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_8"><i><font color="Green" title="E34">A276</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_8"><i><font color="Green" title="E35">A277</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8_1"><i><font color="Green" title="E36">A278</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green" title="E37">A279</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_8_1"><i><font color="Green" title="E38">A280</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8_1"><i><font color="Green" title="E39">A281</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E28:1_1_1"/><i><font color="Green" title="E24">A282</font></i>: 
for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b5">j2</font> &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">i2</font>, <font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_9"/><i><font color="Green" title="E25">A283</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:1_1_1_9"/><i><font color="Green" title="E26">A284</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font> )
 <b>and </b><br/><a NAME="E3:1_1_1_9"/><i><font color="Green" title="E27">A285</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font> )
 ;<br/>

<a NAME="E4:1_1_1_9"/><i><font color="Green" title="E28">A286</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/>
<a NAME="E5:1_1_1_9"/><i><font color="Green" title="E29">A287</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/>
<a NAME="E6:1_1_1_9"/><i><font color="Green" title="E30">A288</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/>
<a NAME="E7:1_1_1_9"/><b>then </b><i><font color="Green" title="E31">A289</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_9"><i><font color="Green" title="E25">A283</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/>
<a NAME="E8:1_1_1_9"/><i><font color="Green" title="E32">A290</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_9"><i><font color="Green" title="E25">A283</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_9"><i><font color="Green" title="E30">A288</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E9:1_1_1_9"/><b>then </b><i><font color="Green" title="E33">A291</font></i>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_9"><i><font color="Green" title="E31">A289</font></i></a></i>;<br/>
<a NAME="E10:1_1_1_9"/><i><font color="Green" title="E34">A292</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_9"><i><font color="Green" title="E25">A283</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_9"><i><font color="Green" title="E28">A286</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_9"><i><font color="Green" title="E30">A288</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_9"><i><font color="Green" title="E31">A289</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:1_1_1_9"/><i><font color="Green" title="E35">A293</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_9"><i><font color="Green" title="E32">A290</font></i></a></i>;<br/>
<a NAME="E12:1_1_1_9"/><i><font color="Green" title="E36">A294</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">i1</font> &amp; <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">j1</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">j2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>assume </b><a NAME="E13:1_1_1_9"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">i</font>, <font color="Maroon" title="c14">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E15:1_1_1_9"/><i><font color="Green" title="E37">A295</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E16:1_1_1_9"/><i><font color="Green" title="E38">A296</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_9"><i><font color="Green" title="E25">A283</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_9"><i><font color="Green" title="E29">A287</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_9"><i><font color="Green" title="E30">A288</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_9"><i><font color="Green" title="E33">A291</font></i></a></i>;<br/>
<a NAME="E17:1_1_1_9"/><i><font color="Green" title="E39">A297</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_9"><i><font color="Green" title="E26">A284</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_9"><i><font color="Green" title="E27">A285</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_9"><i><font color="Green" title="E34">A292</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green" title="E38">A296</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E18:1_1_1_9"/><i><font color="Green" title="E40">A298</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green" title="E38">A296</font></i></a>, <a class="ref" href="finseq_4.html#T82" title="FINSEQ_4:th.82">FINSEQ_4:82</a></i>;<br/>
<b>thus </b><a NAME="E19:1_1_1_9"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1_9"><i><font color="Green" title="E37">A295</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green" title="E38">A296</font></i></a></i>;<br/>

<b>thus </b><a NAME="E20:1_1_1_9"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_9"><i><font color="Green" title="E26">A284</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_9"><i><font color="Green" title="E27">A285</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_9"><i><font color="Green" title="E32">A290</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_9"><i><font color="Green" title="E35">A293</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_9"><i><font color="Green" title="E36">A294</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_9"><i><font color="Green" title="E37">A295</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green" title="E38">A296</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_9"><i><font color="Green" title="E39">A297</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_9"><i><font color="Green" title="E40">A298</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/>

<b>thus </b><a NAME="E21:1_1_1_9"/>
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_9"><i><font color="Green" title="E26">A284</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_9"><i><font color="Green" title="E27">A285</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_9"><i><font color="Green" title="E32">A290</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_9"><i><font color="Green" title="E35">A293</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_9"><i><font color="Green" title="E36">A294</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_9"><i><font color="Green" title="E37">A295</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green" title="E38">A296</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_9"><i><font color="Green" title="E39">A297</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_9"><i><font color="Green" title="E40">A298</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/>

<b>thus </b><a NAME="E22:1_1_1_9"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_9"><i><font color="Green" title="E26">A284</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_9"><i><font color="Green" title="E27">A285</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_9"><i><font color="Green" title="E32">A290</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_9"><i><font color="Green" title="E35">A293</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_9"><i><font color="Green" title="E36">A294</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_9"><i><font color="Green" title="E37">A295</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green" title="E38">A296</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_9"><i><font color="Green" title="E39">A297</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_9"><i><font color="Green" title="E40">A298</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/>

<b>thus </b><a NAME="E23:1_1_1_9"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_9"><i><font color="Green" title="E26">A284</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_9"><i><font color="Green" title="E27">A285</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_9"><i><font color="Green" title="E32">A290</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_9"><i><font color="Green" title="E35">A293</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_9"><i><font color="Green" title="E36">A294</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_9"><i><font color="Green" title="E37">A295</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green" title="E38">A296</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_9"><i><font color="Green" title="E39">A297</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_9"><i><font color="Green" title="E40">A298</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:1_1_1"/><i><font color="Green" title="E25">A299</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:1_1_1_10"/><i><font color="Green" title="E26">A300</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/>
<a NAME="E2:1_1_1_10"/><i><font color="Green" title="E27">A301</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_10_1"/>
( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 or <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_1"/><b>then </b><i><font color="Green" title="E28">A302</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>take </b>
 <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
;<br/><b>take </b>
 <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
;<br/><b>thus </b><a NAME="E3:1_1_1_10_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="jordan11.html#T8" title="JORDAN11:th.8">JORDAN11:8</a></i>;<br/><b>thus </b><a NAME="E4:1_1_1_10_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green" title="E5">A77</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1"><i><font color="Green" title="E20">A246</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_10_1_1"><i><font color="Green" title="E28">A302</font></i></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_2"/><i><font color="Green" title="E28">A303</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>take </b>
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
;<br/><b>take </b>
 <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
;<br/><b>thus </b><a NAME="E2:1_1_1_10_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="jordan11.html#T9" title="JORDAN11:th.9">JORDAN11:9</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_10_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E24:1_1_1"><i><font color="Green" title="E20">A246</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1"><i><font color="Green" title="E21">A247</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_2"><i><font color="Green" title="E28">A303</font></i></a>, <a class="ref" href="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3"/><i><font color="Green" title="E28">A304</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3"/><b>then </b><i><font color="Green" title="E29">A305</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_10"><i><font color="Green" title="E27">A301</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E3:1_1_1_10_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_10"><i><font color="Green" title="E27">A301</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">j1</font>, <font color="Maroon" title="c11">i2</font>, <font color="Maroon" title="c12">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:1_1_1_10_1_3"/><i><font color="Green" title="E30">A306</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> )
 <b>and </b><br/><a NAME="E6:1_1_1_10_1_3"/><i><font color="Green" title="E31">A307</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font> )
 <b>and </b><br/><a NAME="E7:1_1_1_10_1_3"/><i><font color="Green" title="E32">A308</font></i>: 
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_10"><i><font color="Green" title="E26">A300</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_10_1_3"><i><font color="Green" title="E29">A305</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_10_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_10_1_3_1_1"/>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_1"/><i><font color="Green" title="E33">A309</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_10_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_10_1_3"><i><font color="Green" title="E32">A308</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_1_1_1_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_1_1_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green" title="E22">A248</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_1"><i><font color="Green" title="E33">A309</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_1_1_1_2"/>
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_1_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green" title="E22">A248</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_1"><i><font color="Green" title="E33">A309</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_1_1_1_3"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_1_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green" title="E22">A248</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_1"><i><font color="Green" title="E33">A309</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_1_1_1_3"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_1_1_1_4"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_1_1_1_4"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green" title="E22">A248</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_1"><i><font color="Green" title="E33">A309</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_1_1_1_4"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_2"/><i><font color="Green" title="E33">A310</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_10_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_10_1_3"><i><font color="Green" title="E32">A308</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_2_1_1_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_2_1_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green" title="E23">A265</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_2"><i><font color="Green" title="E33">A310</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_2_1_1_2"/>
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_2_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green" title="E23">A265</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_2"><i><font color="Green" title="E33">A310</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_2_1_1_3"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_2_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green" title="E23">A265</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_2"><i><font color="Green" title="E33">A310</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_2_1_1_3"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_2_1_1_4"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_2_1_1_4"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green" title="E23">A265</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_2"><i><font color="Green" title="E33">A310</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_2_1_1_4"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_3"/><i><font color="Green" title="E33">A311</font></i>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_10_1_3_1_1_3_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_10_1_3"><i><font color="Green" title="E32">A308</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_3_1_1_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_3_1_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green" title="E24">A282</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_3"><i><font color="Green" title="E33">A311</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_3_1_1_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_3_1_1_2"/>
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_3_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green" title="E24">A282</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_3"><i><font color="Green" title="E33">A311</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_3_1_1_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_3_1_1_3"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_3_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green" title="E24">A282</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_3"><i><font color="Green" title="E33">A311</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_3_1_1_3"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_3_1_1_4"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3_1_1_3_1_1_4"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green" title="E24">A282</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green" title="E28">A304</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green" title="E30">A306</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green" title="E31">A307</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_3"><i><font color="Green" title="E33">A311</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_3_1_1_4"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_3"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:1_1_1_10_1_3"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">m</font>;<br/>
<a NAME="E30:1_1_1"/><i><font color="Green" title="E26">A312</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_11"/><i><font color="Green" title="E27">A313</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:1_1_1_11"/><b>then </b>
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font>
 
;<br/>
<a NAME="E3:1_1_1_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E4:1_1_1_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T110" title="RELAT_1:th.110">RELAT_1:110</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_1_11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green" title="E5">A77</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_11"><i><font color="Green" title="E27">A313</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:1_1_1"/><i><font color="Green" title="E27">A314</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_12"/><i><font color="Green" title="E28">A315</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">m</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:1_1_1_12"/><i><font color="Green" title="E29">A316</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_12_1"/>
( <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_12"><i><font color="Green" title="E29">A316</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_12_1_1"/>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_12_1_1"/><b>then </b><i><font color="Green" title="E30">A317</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:1_1_1_12_1_1"/><i><font color="Green" title="E31">A318</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:1_1_1_12_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_12_1_1"/><i><font color="Green" title="E32">A319</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1"><i><font color="Green" title="E25">A299</font></i></a></i>;<br/><a NAME="E7:1_1_1_12_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_12_1_1"><i><font color="Green" title="E30">A317</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_12_1_1"><i><font color="Green" title="E31">A318</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_12_1_1"><i><font color="Green" title="E32">A319</font></i></a>, <a class="ref" href="finseq_5.html#T25" title="FINSEQ_5:th.25">FINSEQ_5:25</a></i>;<br/><b>hence </b><a NAME="E8:1_1_1_12_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_12"><i><font color="Green" title="E28">A315</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_12_1_1"><i><font color="Green" title="E30">A317</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_12_1_2"/><i><font color="Green" title="E30">A320</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_12_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_12_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_12_1_2"><i><font color="Green" title="E30">A320</font></i></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E32:1_1_1"/><i><font color="Green" title="E28">A321</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E30:1_1_1"><i><font color="Green" title="E26">A312</font></i></a>, <a class="txt" href="jordan13.html#E31:1_1_1"><i><font color="Green" title="E27">A314</font></i></a>);<br/></i>
<a NAME="E33:1_1_1"/><i><font color="Green" title="E29">A322</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 holds <br/>( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_13"/><i><font color="Green" title="E30">A323</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/>

<a NAME="E2:1_1_1_13"/><i><font color="Green" title="E31">A324</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/>
<a NAME="E3:1_1_1_13"/><i><font color="Green" title="E32">A325</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/>
<a NAME="E4:1_1_1_13"/><b>then </b><i><font color="Green" title="E33">A326</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_13"><i><font color="Green" title="E30">A323</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/>
<a NAME="E5:1_1_1_13"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_13"><i><font color="Green" title="E30">A323</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_13"><i><font color="Green" title="E32">A325</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E6:1_1_1_13"/><b>then </b>
 ex <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font> &amp; ( ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> ) or ( <font color="Olive" title="b1">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> ) or ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> ) or ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) ) )
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_13"><i><font color="Green" title="E31">A324</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_13"><i><font color="Green" title="E33">A326</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_1_13"/>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green" title="E22">A248</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green" title="E23">A265</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green" title="E24">A282</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_13"><i><font color="Green" title="E30">A323</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_13"><i><font color="Green" title="E32">A325</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1 is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>;<br/>
<a NAME="E34:1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green" title="E5">A77</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E35:1_1_1"/><b>then </b><i><font color="Green" title="E30">A327</font></i>: 
<font color="Maroon">S<sub>6</sub></font>[0]
 
<b>by </b><i><a class="ref" href="sppol_2.html#T27" title="SPPOL_2:th.27">SPPOL_2:27</a></i>;<br/>
<a NAME="E36:1_1_1"/><i><font color="Green" title="E31">A328</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_14"/><i><font color="Green" title="E32">A329</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 ;<br/>

<a NAME="E2:1_1_1_14"/><i><font color="Green" title="E33">A330</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_14_1"/>
( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_1"/><b>then </b>
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:1_1_1_14_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_14_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="ref" href="sppol_2.html#T27" title="SPPOL_2:th.27">SPPOL_2:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2"/><i><font color="Green" title="E34">A331</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E2:1_1_1_14_1_2"/><i><font color="Green" title="E35">A332</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E3:1_1_1_14_1_2"/><i><font color="Green" title="E36">A333</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E4:1_1_1_14_1_2"/><i><font color="Green" title="E37">A334</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">i1</font>, <font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">i2</font>, <font color="Maroon" title="c13">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:1_1_1_14_1_2"/><i><font color="Green" title="E38">A335</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">j1</font> )
 <b>and </b><br/><a NAME="E7:1_1_1_14_1_2"/><i><font color="Green" title="E39">A336</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c13">j2</font> )
 <b>and </b><br/><a NAME="E8:1_1_1_14_1_2"/><i><font color="Green" title="E40">A337</font></i>: 
( ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_14"><i><font color="Green" title="E33">A330</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2"><i><font color="Green" title="E35">A332</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E9:1_1_1_14_1_2"/><i><font color="Green" title="E41">A338</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i1</font> &amp; <font color="Maroon" title="c10">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j1</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E10:1_1_1_14_1_2"/><i><font color="Green" title="E42">A339</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c12">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font> &amp; <font color="Maroon" title="c13">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E11:1_1_1_14_1_2"/><b>then </b><i><font color="Green" title="E43">A340</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E12:1_1_1_14_1_2"/><i><font color="Green" title="E44">A341</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green" title="E42">A339</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E13:1_1_1_14_1_2"/><i><font color="Green" title="E45">A342</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c13">j2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_14_1_2"><i><font color="Green" title="E35">A332</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_14_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_14_1_2_1_1"/>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_1"/><i><font color="Green" title="E46">A343</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_14_1_2_1_1_1_1_1"/>
( ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_14_1_2"><i><font color="Green" title="E40">A337</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E47">A344</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E48">A345</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green" title="E22">A248</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E46">A343</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green" title="E42">A339</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_14_1_2"><i><font color="Green" title="E43">A340</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1_1_1_1"><i><font color="Green" title="E47">A344</font></i></a>, <a class="ref" href="goboard7.html#T18" title="GOBOARD7:th.18">GOBOARD7:18</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_1_1_1_1"><i><font color="Green" title="E48">A345</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E47">A346</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E48">A347</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green" title="E22">A248</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E46">A343</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green" title="E42">A339</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1_1_1_2"><i><font color="Green" title="E47">A346</font></i></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_1_1_1_2"><i><font color="Green" title="E48">A347</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_1_1_1_3"/><i><font color="Green" title="E47">A348</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E48">A349</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green" title="E22">A248</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E46">A343</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_1_1_1_3"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green" title="E42">A339</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_14_1_2"><i><font color="Green" title="E44">A341</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1_1_1_3"><i><font color="Green" title="E47">A348</font></i></a>, <a class="ref" href="goboard7.html#T17" title="GOBOARD7:th.17">GOBOARD7:17</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_1_1_1_3"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_1_1_1_3"><i><font color="Green" title="E48">A349</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_1_1_1_4"/><i><font color="Green" title="E47">A350</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_1_1_1_4"/><b>then </b><i><font color="Green" title="E48">A351</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green" title="E22">A248</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1"><i><font color="Green" title="E46">A343</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green" title="E42">A339</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1_1_1_4"><i><font color="Green" title="E47">A350</font></i></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_1_1_1_4"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_1_1_1_4"><i><font color="Green" title="E48">A351</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_14_1_2_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_2"/><i><font color="Green" title="E46">A352</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_14_1_2_1_1_2_1_1"/>
( ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_14_1_2"><i><font color="Green" title="E40">A337</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E47">A353</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E48">A354</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green" title="E23">A265</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2"><i><font color="Green" title="E46">A352</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E47">A353</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E47">A353</font></i></a>, <a class="ref" href="goboard7.html#T15" title="GOBOARD7:th.15">GOBOARD7:15</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green" title="E48">A354</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E47">A355</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E48">A356</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green" title="E23">A265</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2"><i><font color="Green" title="E46">A352</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E47">A355</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E47">A355</font></i></a>, <a class="ref" href="goboard7.html#T16" title="GOBOARD7:th.16">GOBOARD7:16</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green" title="E48">A356</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_2_1_1_3"/><i><font color="Green" title="E47">A357</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E48">A358</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green" title="E23">A265</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2"><i><font color="Green" title="E46">A352</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_2_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_14_1_2"><i><font color="Green" title="E43">A340</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_3"><i><font color="Green" title="E47">A357</font></i></a>, <a class="ref" href="goboard7.html#T16" title="GOBOARD7:th.16">GOBOARD7:16</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_2_1_1_3"><i><font color="Green" title="E48">A358</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_2_1_1_4"/><i><font color="Green" title="E47">A359</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_2_1_1_4"/><b>then </b><i><font color="Green" title="E48">A360</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green" title="E23">A265</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2"><i><font color="Green" title="E46">A352</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_2_1_1_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_2_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_14_1_2"><i><font color="Green" title="E44">A341</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_4"><i><font color="Green" title="E47">A359</font></i></a>, <a class="ref" href="goboard7.html#T15" title="GOBOARD7:th.15">GOBOARD7:15</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_2_1_1_4"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_2_1_1_4"><i><font color="Green" title="E48">A360</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_14_1_2_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_3"/><i><font color="Green" title="E46">A361</font></i>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_14_1_2_1_1_3_1_1"/>
( ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_14_1_2"><i><font color="Green" title="E40">A337</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E47">A362</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E48">A363</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green" title="E24">A282</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3"><i><font color="Green" title="E46">A361</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green" title="E42">A339</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3_1_1_1"><i><font color="Green" title="E47">A362</font></i></a>, <a class="ref" href="goboard7.html#T17" title="GOBOARD7:th.17">GOBOARD7:17</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_3_1_1_1"><i><font color="Green" title="E48">A363</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_3_1_1_2"/><i><font color="Green" title="E47">A364</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E48">A365</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green" title="E24">A282</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3"><i><font color="Green" title="E46">A361</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_3_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green" title="E42">A339</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_14_1_2"><i><font color="Green" title="E44">A341</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3_1_1_2"><i><font color="Green" title="E47">A364</font></i></a>, <a class="ref" href="goboard7.html#T18" title="GOBOARD7:th.18">GOBOARD7:18</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_3_1_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_3_1_1_2"><i><font color="Green" title="E48">A365</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_3_1_1_3"/><i><font color="Green" title="E47">A366</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_3_1_1_3"/><b>then </b><i><font color="Green" title="E48">A367</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green" title="E24">A282</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3"><i><font color="Green" title="E46">A361</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_3_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green" title="E42">A339</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3_1_1_3"><i><font color="Green" title="E47">A366</font></i></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_3_1_1_3"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_3_1_1_3"><i><font color="Green" title="E48">A367</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_3_1_1_4"/><i><font color="Green" title="E47">A368</font></i>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_3_1_1_4"/><b>then </b><i><font color="Green" title="E48">A369</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green" title="E24">A282</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green" title="E34">A331</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green" title="E38">A335</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3"><i><font color="Green" title="E46">A361</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_3_1_1_4"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_1_2_1_1_3_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green" title="E39">A336</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green" title="E41">A338</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green" title="E42">A339</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_14_1_2"><i><font color="Green" title="E43">A340</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green" title="E45">A342</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3_1_1_4"><i><font color="Green" title="E47">A368</font></i></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_3_1_1_4"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green" title="E32">A329</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green" title="E36">A333</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green" title="E37">A334</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_3_1_1_4"><i><font color="Green" title="E48">A369</font></i></a>, <a class="ref" href="sppol_2.html#T31" title="SPPOL_2:th.31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_14_1_2_1_1_3"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:1_1_1_14_1_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E37:1_1_1"/><i><font color="Green" title="E32">A370</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E35:1_1_1"><i><font color="Green" title="E30">A327</font></i></a>, <a class="txt" href="jordan13.html#E36:1_1_1"><i><font color="Green" title="E31">A328</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>7</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span></span>)</span> ) );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_15"/><i><font color="Green" title="E33">A371</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>8</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1 is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>;<br/><a NAME="E2:1_1_1_15"/><i><font color="Green" title="E34">A372</font></i>: 
<font color="Maroon">S<sub>8</sub></font>[0]
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green" title="E5">A77</font></i></a></i>;<br/><a NAME="E3:1_1_1_15"/><i><font color="Green" title="E35">A373</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_15_1"/><i><font color="Green" title="E36">A374</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">n</font>, <font color="Maroon" title="c10">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_15_1_1"/><i><font color="Green" title="E37">A375</font></i>: 
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_15_1_1"/><i><font color="Green" title="E38">A376</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">m</font>
 ;<br/><b>consider </b><font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c12">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:1_1_1_15_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E5:1_1_1_15_1_1"/><i><font color="Green" title="E39">A377</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1"><i><font color="Green" title="E25">A299</font></i></a></i>;<br/><a NAME="E6:1_1_1_15_1_1"/><i><font color="Green" title="E40">A378</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_15_1_1"><i><font color="Green" title="E37">A375</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:1_1_1_15_1_1"/><i><font color="Green" title="E41">A379</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/><a NAME="E8:1_1_1_15_1_1"/><i><font color="Green" title="E42">A380</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_15_1_1_1"/>
( ( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> ) or ( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> ) or ( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_15_1_1"><i><font color="Green" title="E40">A378</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_15_1_1"><i><font color="Green" title="E41">A379</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_15_1_1_1_1"/>
( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_15_1_1_1_1"/><b>then </b><i><font color="Green" title="E43">A381</font></i>: 
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> &amp; <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_15_1_1"><i><font color="Green" title="E40">A378</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_15_1_1"><i><font color="Green" title="E41">A379</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_1_15_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">m</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_15_1_1"><i><font color="Green" title="E39">A377</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_15_1_1_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_15_1"><i><font color="Green" title="E36">A374</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_15_1_1"><i><font color="Green" title="E38">A376</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_15_1_1_1_1"><i><font color="Green" title="E43">A381</font></i></a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_15_1_1_1_2"/>
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_15_1_1_1_2"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_15"><i><font color="Green" title="E33">A371</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_15_1_1"><i><font color="Green" title="E37">A375</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_15_1_1"><i><font color="Green" title="E38">A376</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_15_1_1"><i><font color="Green" title="E41">A379</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_15_1_1"><i><font color="Green" title="E42">A380</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_15_1_1_1_3"/>
( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_15_1_1_1_3"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_15"><i><font color="Green" title="E33">A371</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_15_1_1"><i><font color="Green" title="E37">A375</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_15_1_1"><i><font color="Green" title="E38">A376</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_15_1_1"><i><font color="Green" title="E41">A379</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_15_1_1"><i><font color="Green" title="E42">A380</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_15_1_1_1_4"/>
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_15_1_1_1_4"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:1_1_1_15_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="partfun2.html#T16" title="PARTFUN2:th.16">PARTFUN2:16</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:1_1_1_15"/><i><font color="Green" title="E36">A382</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E2:1_1_1_15"><i><font color="Green" title="E34">A372</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_15"><i><font color="Green" title="E35">A373</font></i></a>);<br/></i><a NAME="E5:1_1_1_15"/><i><font color="Green" title="E37">A383</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:1_1_1_15_2"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_15"><i><font color="Green" title="E36">A382</font></i></a></i>;<br/>
<b>hence </b> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>
<br/>.= 
<font color="Maroon" title="c8">k</font>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c8">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E7:1_1_1_15"/><i><font color="Green" title="E38">A384</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:1_1_1_15"/><i><font color="Green" title="E39">A385</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="gobrd13.html#T8" title="GOBRD13:th.8">GOBRD13:8</a></i>;<br/><a NAME="E9:1_1_1_15"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/><a NAME="E10:1_1_1_15"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="gobrd13.html#T9" title="GOBRD13:th.9">GOBRD13:9</a></i>;<br/><a NAME="E11:1_1_1_15"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T80" title="CARD_1:th.80">CARD_1:80</a></i>;<br/><a NAME="E12:1_1_1_15"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_15"><i><font color="Green" title="E39">A385</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E13:1_1_1_15"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_15"><i><font color="Green" title="E37">A383</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_15"><i><font color="Green" title="E38">A384</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E39:1_1_1"/><b>then </b><i><font color="Green" title="E33">A386</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c8">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E41:1_1_1"/><i><font color="Green" title="E34">A387</font></i>: 
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font>]
 <b>and </b><br/><a NAME="E42:1_1_1"/><i><font color="Green" title="E35">A388</font></i>: 
for <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">l</font>] holds <br/><font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="jordan13.html#E39:1_1_1"><i><font color="Green" title="E33">A386</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c9">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E44:1_1_1"/><i><font color="Green" title="E36">A389</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>and </b><br/><a NAME="E45:1_1_1"/><i><font color="Green" title="E37">A390</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <b>and </b><br/><a NAME="E46:1_1_1"/><i><font color="Green" title="E38">A391</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E41:1_1_1"><i><font color="Green" title="E34">A387</font></i></a></i>;<br/>
<a NAME="E47:1_1_1"/><i><font color="Green" title="E39">A392</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan13.html#E41:1_1_1"><i><font color="Green" title="E34">A387</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E49:1_1_1"/><i><font color="Green" title="E40">A393</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/>
<a NAME="E50:1_1_1"/><i><font color="Green" title="E41">A394</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E44:1_1_1"><i><font color="Green" title="E36">A389</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E51:1_1_1"/><b>then </b><i><font color="Green" title="E42">A395</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E45:1_1_1"><i><font color="Green" title="E37">A390</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E52:1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E53:1_1_1"/><b>then </b><i><font color="Green" title="E43">A396</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E54:1_1_1"/><b>then </b><i><font color="Green" title="E44">A397</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">f</font> as  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1"><i><font color="Green" title="E32">A370</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E53:1_1_1"><i><font color="Green" title="E43">A396</font></i></a>, <a class="ref" href="jordan8.html#T7" title="JORDAN8:th.7">JORDAN8:7</a>, <a class="ref" href="jordan8.html#T8" title="JORDAN8:th.8">JORDAN8:8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c11">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>;<br/>
<a NAME="E56:1_1_1"/><i><font color="Green" title="E45">A398</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green" title="E42">A395</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E57:1_1_1"/>
( <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E58:1_1_1"/><b>then </b>
<font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E59:1_1_1"/><b>then </b><i><font color="Green" title="E46">A399</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E60:1_1_1"/><b>then </b><i><font color="Green" title="E47">A400</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E61:1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c11">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E63:1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<b>then </b><i><font color="Green" title="E48">A401</font></i>: <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">m</font>
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>
;<br/>
<a NAME="E65:1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<b>then </b><i><font color="Green" title="E49">A402</font></i>: <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font></span>)</span> = 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green" title="E47">A400</font></i></a></i>
;<br/>
<a NAME="E67:1_1_1"/><i><font color="Green" title="E50">A403</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="ref" href="jordan8.html#T5" title="JORDAN8:th.5">JORDAN8:5</a></i>;<br/>
<a NAME="E68:1_1_1"/><b>then </b><i><font color="Green" title="E51">A404</font></i>: 
<font color="Maroon" title="c13">g</font> is <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>
 
<b>by </b><i><a class="ref" href="jordan8.html#T7" title="JORDAN8:th.7">JORDAN8:7</a></i>;<br/>
<a NAME="E69:1_1_1"/><i><font color="Green" title="E52">A405</font></i>: 
for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> &amp; <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">j</font>, <font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_18"/><i><font color="Green" title="E53">A406</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> )
 <b>and </b><br/><a NAME="E2:1_1_1_18"/><i><font color="Green" title="E54">A407</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> )
 <b>and </b><br/><a NAME="E3:1_1_1_18"/><i><font color="Green" title="E55">A408</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">j</font>
 <b>and </b><br/><a NAME="E4:1_1_1_18"/><i><font color="Green" title="E56">A409</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">j</font>
 ;<br/>

<a NAME="E5:1_1_1_18"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_18"><i><font color="Green" title="E53">A406</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:1_1_1_18"/><b>then </b><i><font color="Green" title="E57">A410</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>;<br/>
<a NAME="E7:1_1_1_18"/>
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_18"><i><font color="Green" title="E54">A407</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:1_1_1_18"/><b>then </b><i><font color="Green" title="E58">A411</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_18_1"/>
( <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">j</font> or <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">i</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_18"><i><font color="Green" title="E56">A409</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_18_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_18_1_1"/><i><font color="Green" title="E59">A412</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">j</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c16">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font>;<br/><b>set </b><font color="Maroon" title="c17">m'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font>;<br/><a NAME="E2:1_1_1_18_1_1"/><i><font color="Green" title="E60">A413</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_18_1_1"><i><font color="Green" title="E59">A412</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:1_1_1_18_1_1"/><i><font color="Green" title="E61">A414</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green" title="E47">A400</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_18"><i><font color="Green" title="E54">A407</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E4:1_1_1_18_1_1"/><b>then </b><i><font color="Green" title="E62">A415</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green" title="E28">A321</font></i></a></i>;<br/><a NAME="E5:1_1_1_18_1_1"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:1_1_1_18_1_1"/><b>then </b><i><font color="Green" title="E63">A416</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_18"><i><font color="Green" title="E54">A407</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:1_1_1_18_1_1"/><i><font color="Green" title="E64">A417</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/><a NAME="E8:1_1_1_18_1_1"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E9:1_1_1_18_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_18"><i><font color="Green" title="E53">A406</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:1_1_1_18_1_1"/><b>then </b><i><font color="Green" title="E65">A418</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_18_1_1"><i><font color="Green" title="E60">A413</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_1"><i><font color="Green" title="E64">A417</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:1_1_1_18_1_1"/><b>then </b><i><font color="Green" title="E66">A419</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_18_1_1"><i><font color="Green" title="E62">A415</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><a NAME="E12:1_1_1_18_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_18_1_1"><i><font color="Green" title="E63">A416</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_1"><i><font color="Green" title="E64">A417</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:1_1_1_18_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_18_1_1"><i><font color="Green" title="E62">A415</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><b>hence </b><a NAME="E14:1_1_1_18_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green" title="E35">A388</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_18"><i><font color="Green" title="E55">A408</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_18"><i><font color="Green" title="E57">A410</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_18"><i><font color="Green" title="E58">A411</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_18_1_1"><i><font color="Green" title="E60">A413</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_18_1_1"><i><font color="Green" title="E61">A414</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_18_1_1"><i><font color="Green" title="E63">A416</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_1"><i><font color="Green" title="E64">A417</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_18_1_1"><i><font color="Green" title="E65">A418</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_18_1_1"><i><font color="Green" title="E66">A419</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_18_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_18_1_2"/><i><font color="Green" title="E59">A420</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">i</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c16">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font>;<br/><b>set </b><font color="Maroon" title="c17">m'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font>;<br/><a NAME="E2:1_1_1_18_1_2"/><i><font color="Green" title="E60">A421</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_18_1_2"><i><font color="Green" title="E59">A420</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:1_1_1_18_1_2"/><i><font color="Green" title="E61">A422</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green" title="E47">A400</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_18"><i><font color="Green" title="E53">A406</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E4:1_1_1_18_1_2"/><b>then </b><i><font color="Green" title="E62">A423</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green" title="E28">A321</font></i></a></i>;<br/><a NAME="E5:1_1_1_18_1_2"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:1_1_1_18_1_2"/><b>then </b><i><font color="Green" title="E63">A424</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_18"><i><font color="Green" title="E53">A406</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:1_1_1_18_1_2"/><i><font color="Green" title="E64">A425</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/><a NAME="E8:1_1_1_18_1_2"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E9:1_1_1_18_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_18"><i><font color="Green" title="E54">A407</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:1_1_1_18_1_2"/><b>then </b><i><font color="Green" title="E65">A426</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_18_1_2"><i><font color="Green" title="E60">A421</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_2"><i><font color="Green" title="E64">A425</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:1_1_1_18_1_2"/><b>then </b><i><font color="Green" title="E66">A427</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c14">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_18_1_2"><i><font color="Green" title="E62">A423</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><a NAME="E12:1_1_1_18_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_18_1_2"><i><font color="Green" title="E63">A424</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_2"><i><font color="Green" title="E64">A425</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:1_1_1_18_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_18_1_2"><i><font color="Green" title="E62">A423</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><b>hence </b><a NAME="E14:1_1_1_18_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green" title="E35">A388</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_18"><i><font color="Green" title="E55">A408</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_18"><i><font color="Green" title="E57">A410</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_18"><i><font color="Green" title="E58">A411</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_18_1_2"><i><font color="Green" title="E60">A421</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_18_1_2"><i><font color="Green" title="E61">A422</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_18_1_2"><i><font color="Green" title="E63">A424</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_2"><i><font color="Green" title="E64">A425</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_18_1_2"><i><font color="Green" title="E65">A426</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_18_1_2"><i><font color="Green" title="E66">A427</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E70:1_1_1"/><i><font color="Green" title="E53">A428</font></i>: 
for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> holds <br/><font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">j</font>, <font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_19"/><i><font color="Green" title="E54">A429</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">i</font>
 <b>and </b><br/><a NAME="E2:1_1_1_19"/><i><font color="Green" title="E55">A430</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">j</font>
 <b>and </b><br/><a NAME="E3:1_1_1_19"/><i><font color="Green" title="E56">A431</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>and </b><br/><a NAME="E4:1_1_1_19"/><i><font color="Green" title="E57">A432</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">j</font>
 ;<br/>

<a NAME="E5:1_1_1_19"/><i><font color="Green" title="E58">A433</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_19"><i><font color="Green" title="E54">A429</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_19"><i><font color="Green" title="E55">A430</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:1_1_1_19"/><i><font color="Green" title="E59">A434</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_19"><i><font color="Green" title="E55">A430</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_19"><i><font color="Green" title="E56">A431</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:1_1_1_19"/><b>then </b><i><font color="Green" title="E60">A435</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_19"><i><font color="Green" title="E54">A429</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_19_1"/>
( <font color="Maroon" title="c14">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> or <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_19_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_19_1_1"/>
<font color="Maroon" title="c14">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_19_1_1"/><b>then </b>
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_19"><i><font color="Green" title="E56">A431</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_19_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_19"><i><font color="Green" title="E54">A429</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_19"><i><font color="Green" title="E55">A430</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_19"><i><font color="Green" title="E57">A432</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_19"><i><font color="Green" title="E58">A433</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_19"><i><font color="Green" title="E59">A434</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_19_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_19_1_2"/>
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_19_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green" title="E38">A391</font></i></a>, <a class="txt" href="jordan13.html#E64:1_1_1"><i><font color="Green" title="E48">A401</font></i></a>, <a class="txt" href="jordan13.html#E66:1_1_1"><i><font color="Green" title="E49">A402</font></i></a>, <a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_19"><i><font color="Green" title="E54">A429</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_19"><i><font color="Green" title="E55">A430</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_19"><i><font color="Green" title="E57">A432</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_19"><i><font color="Green" title="E60">A435</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E71:1_1_1"/><i><font color="Green" title="E54">A436</font></i>: 
for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> holds <br/><font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">j</font>, <font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_20"/><i><font color="Green" title="E55">A437</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font>
 <b>and </b><br/><a NAME="E2:1_1_1_20"/><i><font color="Green" title="E56">A438</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">j</font>
 <b>and </b><br/><a NAME="E3:1_1_1_20"/><i><font color="Green" title="E57">A439</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>and </b><br/><a NAME="E4:1_1_1_20"/><i><font color="Green" title="E58">A440</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">j</font>
 ;<br/>

<a NAME="E5:1_1_1_20"/><i><font color="Green" title="E59">A441</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">j</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_20"><i><font color="Green" title="E55">A437</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_20"><i><font color="Green" title="E56">A438</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:1_1_1_20"/>
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_20"><i><font color="Green" title="E56">A438</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_20"><i><font color="Green" title="E57">A439</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_1_20"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_20"><i><font color="Green" title="E55">A437</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_20"><i><font color="Green" title="E56">A438</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_20"><i><font color="Green" title="E57">A439</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_20"><i><font color="Green" title="E58">A440</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_20"><i><font color="Green" title="E59">A441</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E72:1_1_1"/><i><font color="Green" title="E55">A442</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> holds <br/> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Olive" title="b1">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Olive" title="b1">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_21"/><i><font color="Green" title="E56">A443</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c15">i1</font>, <font color="Maroon" title="c16">j1</font>, <font color="Maroon" title="c17">i2</font>, <font color="Maroon" title="c18">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_21"/><i><font color="Green" title="E57">A444</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i1</font>,<font color="Maroon" title="c16">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_21"/><i><font color="Green" title="E58">A445</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1</font>,<font color="Maroon" title="c16">j1</font>
 <b>and </b><br/><a NAME="E5:1_1_1_21"/><i><font color="Green" title="E59">A446</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">i2</font>,<font color="Maroon" title="c18">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_21"/><i><font color="Green" title="E60">A447</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c17">i2</font>,<font color="Maroon" title="c18">j2</font>
 <b>and </b><br/><a NAME="E7:1_1_1_21"/><i><font color="Green" title="E61">A448</font></i>: 
( ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21"><i><font color="Green" title="E56">A443</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/>
<a NAME="E8:1_1_1_21"/><i><font color="Green" title="E62">A449</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i1</font> &amp; <font color="Maroon" title="c15">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_21"><i><font color="Green" title="E57">A444</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E9:1_1_1_21"/><i><font color="Green" title="E63">A450</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c17">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j2</font> &amp; <font color="Maroon" title="c18">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_21"><i><font color="Green" title="E59">A446</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E10:1_1_1_21"/><i><font color="Green" title="E64">A451</font></i>: 
( <font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">i1</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">j1</font> &amp; <font color="Maroon" title="c17">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c18">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">j2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_21_1"/>
( ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_21"><i><font color="Green" title="E61">A448</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_21_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_21_1_1"/><i><font color="Green" title="E65">A452</font></i>: 
( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_21_1_1"/><i><font color="Green" title="E66">A453</font></i>: 
<font color="Maroon" title="c15">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_21"><i><font color="Green" title="E62">A449</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E3:1_1_1_21_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j1</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21"><i><font color="Green" title="E56">A443</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_21"><i><font color="Green" title="E57">A444</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_21"><i><font color="Green" title="E58">A445</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_21"><i><font color="Green" title="E59">A446</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_21"><i><font color="Green" title="E60">A447</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_21"><i><font color="Green" title="E64">A451</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21_1_1"><i><font color="Green" title="E65">A452</font></i></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_21_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_21"><i><font color="Green" title="E62">A449</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_21_1_1"><i><font color="Green" title="E66">A453</font></i></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_21_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_21_1_2"/>
( <font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_21_1_2"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i1</font>,<font color="Maroon" title="c16">j1</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21"><i><font color="Green" title="E56">A443</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_21"><i><font color="Green" title="E57">A444</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_21"><i><font color="Green" title="E58">A445</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_21"><i><font color="Green" title="E59">A446</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_21"><i><font color="Green" title="E60">A447</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_21"><i><font color="Green" title="E64">A451</font></i></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_21_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_21"><i><font color="Green" title="E62">A449</font></i></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_21_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_21_1_3"/><i><font color="Green" title="E65">A454</font></i>: 
( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_21_1_3"/><i><font color="Green" title="E66">A455</font></i>: 
<font color="Maroon" title="c18">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_21"><i><font color="Green" title="E63">A450</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E3:1_1_1_21_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c17">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21"><i><font color="Green" title="E56">A443</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_21"><i><font color="Green" title="E57">A444</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_21"><i><font color="Green" title="E58">A445</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_21"><i><font color="Green" title="E59">A446</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_21"><i><font color="Green" title="E60">A447</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_21"><i><font color="Green" title="E64">A451</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21_1_3"><i><font color="Green" title="E65">A454</font></i></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_21_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_21"><i><font color="Green" title="E63">A450</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_21_1_3"><i><font color="Green" title="E66">A455</font></i></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_21_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_21_1_4"/>
( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_21_1_4"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c15">i1</font>,<font color="Maroon" title="c18">j2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21"><i><font color="Green" title="E56">A443</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_21"><i><font color="Green" title="E57">A444</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_21"><i><font color="Green" title="E58">A445</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_21"><i><font color="Green" title="E59">A446</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_21"><i><font color="Green" title="E60">A447</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_21"><i><font color="Green" title="E64">A451</font></i></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_21_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_21"><i><font color="Green" title="E62">A449</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_21"><i><font color="Green" title="E63">A450</font></i></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E73:1_1_1"/><i><font color="Green" title="E56">A456</font></i>: 
<font color="Maroon" title="c13">g</font> is <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard5.html#D4" title="GOBOARD5:def.4">GOBOARD5:def 4</a><br/><b>let </b><font color="Maroon" title="c15">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_22"/><i><font color="Green" title="E57">A457</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">j</font>
 <b>and </b><br/><a NAME="E2:1_1_1_22"/><i><font color="Green" title="E58">A458</font></i>: 
( ( <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) or <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> )
 ;<br/>

<a NAME="E3:1_1_1_22"/><i><font color="Green" title="E59">A459</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E4:1_1_1_22"/><b>then </b><i><font color="Green" title="E60">A460</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:1_1_1_22"/><i><font color="Green" title="E61">A461</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E6:1_1_1_22"/><i><font color="Green" title="E62">A462</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">j</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E7:1_1_1_22"/>
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">j</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:1_1_1_22"/><b>then </b><i><font color="Green" title="E63">A463</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1"/>
( ( <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) or ( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) or ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_22"><i><font color="Green" title="E58">A458</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1"/><i><font color="Green" title="E64">A464</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1"/><b>then </b><i><font color="Green" title="E65">A465</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:1_1_1_22_1_1"/><b>then </b><i><font color="Green" title="E66">A466</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:1_1_1_22_1_1"/><i><font color="Green" title="E67">A467</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:1_1_1_22_1_1"/><i><font color="Green" title="E68">A468</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">i1</font>, <font color="Maroon" title="c17">j1</font>, <font color="Maroon" title="c18">i2</font>, <font color="Maroon" title="c19">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:1_1_1_22_1_1"/><i><font color="Green" title="E69">A469</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E8:1_1_1_22_1_1"/><i><font color="Green" title="E70">A470</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font>
 <b>and </b><br/><a NAME="E9:1_1_1_22_1_1"/><i><font color="Green" title="E71">A471</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2</font>,<font color="Maroon" title="c19">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E10:1_1_1_22_1_1"/><i><font color="Green" title="E72">A472</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2</font>,<font color="Maroon" title="c19">j2</font>
 <b>and </b><br/><a NAME="E11:1_1_1_22_1_1"/><i><font color="Green" title="E73">A473</font></i>: 
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green" title="E50">A403</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E12:1_1_1_22_1_1"/><i><font color="Green" title="E74">A474</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c16">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">j1</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_22_1_1"><i><font color="Green" title="E69">A469</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E13:1_1_1_22_1_1"/><i><font color="Green" title="E75">A475</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c18">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_22_1_1"><i><font color="Green" title="E71">A471</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E14:1_1_1_22_1_1"/><i><font color="Green" title="E76">A476</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E15:1_1_1_22_1_1"/><b>then </b><i><font color="Green" title="E77">A477</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">i1'</font>, <font color="Maroon" title="c21">j1'</font>, <font color="Maroon" title="c22">i2'</font>, <font color="Maroon" title="c23">j2'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E17:1_1_1_22_1_1"/><i><font color="Green" title="E78">A478</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i1'</font>,<font color="Maroon" title="c21">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E18:1_1_1_22_1_1"/><i><font color="Green" title="E79">A479</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i1'</font>,<font color="Maroon" title="c21">j1'</font>
 <b>and </b><br/><a NAME="E19:1_1_1_22_1_1"/><i><font color="Green" title="E80">A480</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">i2'</font>,<font color="Maroon" title="c23">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E20:1_1_1_22_1_1"/><i><font color="Green" title="E81">A481</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">i2'</font>,<font color="Maroon" title="c23">j2'</font>
 <b>and </b><br/><a NAME="E21:1_1_1_22_1_1"/><i><font color="Green" title="E82">A482</font></i>: 
( ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green" title="E50">A403</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E22:1_1_1_22_1_1"/><i><font color="Green" title="E83">A483</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">j1'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1_22_1_1"><i><font color="Green" title="E78">A478</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E23:1_1_1_22_1_1"/><i><font color="Green" title="E84">A484</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c22">i2'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j2'</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E19:1_1_1_22_1_1"><i><font color="Green" title="E80">A480</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>assume </b><a NAME="E24:1_1_1_22_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/><a NAME="E25:1_1_1_22_1_1"/><b>then </b><i><font color="Green" title="E85">A485</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c14">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_22_1_1"><i><font color="Green" title="E73">A473</font></i></a>, <a class="txt" href="jordan13.html#E21:1_1_1_22_1_1"><i><font color="Green" title="E82">A482</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_1"/><i><font color="Green" title="E86">A486</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E87">A487</font></i>: 
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> or <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_1"><i><font color="Green" title="E86">A486</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_1"><i><font color="Green" title="E87">A487</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_1"><i><font color="Green" title="E86">A486</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_1"><i><font color="Green" title="E87">A487</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_1_1_1_3"/>
<font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_1"><i><font color="Green" title="E86">A486</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_1"><i><font color="Green" title="E87">A487</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_2"/><i><font color="Green" title="E86">A488</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_2"><i><font color="Green" title="E86">A488</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_2_1_1_2"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_2"><i><font color="Green" title="E86">A488</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_2_1_1_3"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_2"><i><font color="Green" title="E86">A488</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_2_1_1_4"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_2"><i><font color="Green" title="E86">A488</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_3"/><i><font color="Green" title="E86">A489</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_3"><i><font color="Green" title="E86">A489</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_3_1_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_3"><i><font color="Green" title="E86">A489</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_3_1_1_2"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_3"><i><font color="Green" title="E86">A489</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_3_1_1_3"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_3"><i><font color="Green" title="E86">A489</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_3_1_1_4"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_3_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_3"><i><font color="Green" title="E86">A489</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_4"/><i><font color="Green" title="E86">A490</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_4"/><b>then </b><i><font color="Green" title="E87">A491</font></i>: 
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_4_1_1"/>
( <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> or <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_4"><i><font color="Green" title="E86">A490</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_4_1_1_1"/>
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_4"><i><font color="Green" title="E86">A490</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_4"><i><font color="Green" title="E87">A491</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_4_1_1_2"/>
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_4"><i><font color="Green" title="E86">A490</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_4"><i><font color="Green" title="E87">A491</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_4_1_1_3"/>
<font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_4_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_4"><i><font color="Green" title="E86">A490</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_4"><i><font color="Green" title="E87">A491</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_4"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_5"/><i><font color="Green" title="E86">A492</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_5_1_1"/>
( ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_5"><i><font color="Green" title="E86">A492</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_5_1_1_1"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_5_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_5_1_1_2"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_5_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_5"><i><font color="Green" title="E86">A492</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_5_1_1_3"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_5_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_5"><i><font color="Green" title="E86">A492</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_5_1_1_4"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_5_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_5"><i><font color="Green" title="E86">A492</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_5"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_6"/><i><font color="Green" title="E86">A493</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_6"/><b>then </b><i><font color="Green" title="E87">A494</font></i>: 
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_6_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> or <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_6"><i><font color="Green" title="E86">A493</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_6_1_1_1"/>
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_6_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_6"><i><font color="Green" title="E87">A494</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_6_1_1_2"/>
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_6_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_6"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_6"><i><font color="Green" title="E87">A494</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_6_1_1_3"/>
<font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_6_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_6"><i><font color="Green" title="E86">A493</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_6"><i><font color="Green" title="E87">A494</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_6"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_7"/><i><font color="Green" title="E86">A495</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_7"/><b>then </b><i><font color="Green" title="E87">A496</font></i>: 
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_7_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> or <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_7"><i><font color="Green" title="E86">A495</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_7_1_1_1"/>
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_7_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_7"><i><font color="Green" title="E86">A495</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_7"><i><font color="Green" title="E87">A496</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_7_1_1_2"/>
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_7_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_7"><i><font color="Green" title="E86">A495</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_7"><i><font color="Green" title="E87">A496</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_7_1_1_3"/>
<font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_7_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_7"><i><font color="Green" title="E86">A495</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_7"><i><font color="Green" title="E87">A496</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_7"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_8"/><i><font color="Green" title="E86">A497</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_8_1_1"/>
( ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_8"><i><font color="Green" title="E86">A497</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_8_1_1_1"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_8_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_8"><i><font color="Green" title="E86">A497</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_8_1_1_2"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_8_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_8"><i><font color="Green" title="E86">A497</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_8_1_1_3"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_8_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_8"><i><font color="Green" title="E86">A497</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_8_1_1_4"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_8_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_8"><i><font color="Green" title="E86">A497</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_8"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_9"/><i><font color="Green" title="E86">A498</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_9_1_1"/>
( ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_9"><i><font color="Green" title="E86">A498</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_9_1_1_1"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_9_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_9"><i><font color="Green" title="E86">A498</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_9_1_1_2"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_9_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_9"><i><font color="Green" title="E86">A498</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_9_1_1_3"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_9_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_9"><i><font color="Green" title="E86">A498</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_9_1_1_4"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_9_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_9"><i><font color="Green" title="E86">A498</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_9"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_10"/><i><font color="Green" title="E86">A499</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_10"/><b>then </b><i><font color="Green" title="E87">A500</font></i>: 
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_10_1_1"/>
( <font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> or <font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_10"><i><font color="Green" title="E86">A499</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_10_1_1_1"/>
<font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_10_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_10"><i><font color="Green" title="E86">A499</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_10"><i><font color="Green" title="E87">A500</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_10_1_1_2"/>
<font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_10_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_10"><i><font color="Green" title="E86">A499</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_10"><i><font color="Green" title="E87">A500</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_10_1_1_3"/>
<font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_10_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_10"><i><font color="Green" title="E86">A499</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_10"><i><font color="Green" title="E87">A500</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_10"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_11"/><i><font color="Green" title="E86">A501</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_11"/><b>then </b><i><font color="Green" title="E87">A502</font></i>: 
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_11_1_1"/>
( <font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> or <font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_11"><i><font color="Green" title="E86">A501</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_11_1_1_1"/>
<font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_11_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_11"><i><font color="Green" title="E86">A501</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_11"><i><font color="Green" title="E87">A502</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_11_1_1_2"/>
<font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_11_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_11"><i><font color="Green" title="E86">A501</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_11"><i><font color="Green" title="E87">A502</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_11_1_1_3"/>
<font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_11_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_11"><i><font color="Green" title="E86">A501</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_11"><i><font color="Green" title="E87">A502</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_11"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_12"/><i><font color="Green" title="E86">A503</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_12_1_1"/>
( ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_12"><i><font color="Green" title="E86">A503</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_12_1_1_1"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_12_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_12"><i><font color="Green" title="E86">A503</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_12_1_1_2"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_12_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_12"><i><font color="Green" title="E86">A503</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_12_1_1_3"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_12_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_12"><i><font color="Green" title="E86">A503</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_12_1_1_4"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_12_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_12"><i><font color="Green" title="E86">A503</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_12"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_13"/><i><font color="Green" title="E86">A504</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_13"/><b>then </b><i><font color="Green" title="E87">A505</font></i>: 
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_13_1_1"/>
( <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> or <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_13"><i><font color="Green" title="E86">A504</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_13_1_1_1"/>
<font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_13_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_13"><i><font color="Green" title="E86">A504</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_13"><i><font color="Green" title="E87">A505</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_13_1_1_2"/>
<font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_13_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_13"><i><font color="Green" title="E86">A504</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_13"><i><font color="Green" title="E87">A505</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_13_1_1_3"/>
<font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_13_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_13"><i><font color="Green" title="E86">A504</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_13"><i><font color="Green" title="E87">A505</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_13"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_14"/><i><font color="Green" title="E86">A506</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_14_1_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_14"><i><font color="Green" title="E86">A506</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_14_1_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_14_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_14"><i><font color="Green" title="E86">A506</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_14_1_1_2"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_14_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_14"><i><font color="Green" title="E86">A506</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_14_1_1_3"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_14_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_14"><i><font color="Green" title="E86">A506</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_14_1_1_4"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_14_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_14"><i><font color="Green" title="E86">A506</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_14"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_15"/><i><font color="Green" title="E86">A507</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_15_1_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_15"><i><font color="Green" title="E86">A507</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_15_1_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_15_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_15"><i><font color="Green" title="E86">A507</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_15_1_1_2"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_15_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_15"><i><font color="Green" title="E86">A507</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_15_1_1_3"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_15_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_15"><i><font color="Green" title="E86">A507</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_15_1_1_4"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_15_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green" title="E68">A468</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_15"><i><font color="Green" title="E86">A507</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_15"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_16"/><i><font color="Green" title="E86">A508</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_16"/><b>then </b><i><font color="Green" title="E87">A509</font></i>: 
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_16_1_1"/>
( <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> or <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green" title="E66">A466</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green" title="E74">A474</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green" title="E75">A475</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green" title="E77">A477</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green" title="E83">A483</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green" title="E84">A484</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green" title="E85">A485</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_16"><i><font color="Green" title="E86">A508</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_16_1_1_1"/>
<font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_16_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green" title="E67">A467</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_16"><i><font color="Green" title="E86">A508</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_16"><i><font color="Green" title="E87">A509</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_16_1_1_2"/>
<font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_16_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green" title="E65">A465</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green" title="E72">A472</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green" title="E79">A479</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_16"><i><font color="Green" title="E86">A508</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_16"><i><font color="Green" title="E87">A509</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_16_1_1_3"/>
<font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_16_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green" title="E64">A464</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green" title="E70">A470</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green" title="E76">A476</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green" title="E81">A481</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_16"><i><font color="Green" title="E86">A508</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_16"><i><font color="Green" title="E87">A509</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_16"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E27:1_1_1_22_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_2"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_2"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_22_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3"/><i><font color="Green" title="E64">A510</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3"/><b>then </b><i><font color="Green" title="E65">A511</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:1_1_1_22_1_3"/><b>then </b><i><font color="Green" title="E66">A512</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:1_1_1_22_1_3"/><i><font color="Green" title="E67">A513</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:1_1_1_22_1_3"/><i><font color="Green" title="E68">A514</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:1_1_1_22_1_3"/><i><font color="Green" title="E69">A515</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">i1</font>, <font color="Maroon" title="c17">j1</font>, <font color="Maroon" title="c18">i2</font>, <font color="Maroon" title="c19">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:1_1_1_22_1_3"/><i><font color="Green" title="E70">A516</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E9:1_1_1_22_1_3"/><i><font color="Green" title="E71">A517</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">i1</font>,<font color="Maroon" title="c17">j1</font>
 <b>and </b><br/><a NAME="E10:1_1_1_22_1_3"/><i><font color="Green" title="E72">A518</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i2</font>,<font color="Maroon" title="c19">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E11:1_1_1_22_1_3"/><i><font color="Green" title="E73">A519</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c18">i2</font>,<font color="Maroon" title="c19">j2</font>
 <b>and </b><br/><a NAME="E12:1_1_1_22_1_3"/><i><font color="Green" title="E74">A520</font></i>: 
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green" title="E50">A403</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E13:1_1_1_22_1_3"/><i><font color="Green" title="E75">A521</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c16">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">j1</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_22_1_3"><i><font color="Green" title="E70">A516</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E14:1_1_1_22_1_3"/><i><font color="Green" title="E76">A522</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c18">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_22_1_3"><i><font color="Green" title="E72">A518</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E15:1_1_1_22_1_3"/><i><font color="Green" title="E77">A523</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">i1'</font>, <font color="Maroon" title="c21">j1'</font>, <font color="Maroon" title="c22">i2'</font>, <font color="Maroon" title="c23">j2'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E17:1_1_1_22_1_3"/><i><font color="Green" title="E78">A524</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">i1'</font>,<font color="Maroon" title="c21">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E18:1_1_1_22_1_3"/><i><font color="Green" title="E79">A525</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c20">i1'</font>,<font color="Maroon" title="c21">j1'</font>
 <b>and </b><br/><a NAME="E19:1_1_1_22_1_3"/><i><font color="Green" title="E80">A526</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">i2'</font>,<font color="Maroon" title="c23">j2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E20:1_1_1_22_1_3"/><i><font color="Green" title="E81">A527</font></i>: 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c22">i2'</font>,<font color="Maroon" title="c23">j2'</font>
 <b>and </b><br/><a NAME="E21:1_1_1_22_1_3"/><i><font color="Green" title="E82">A528</font></i>: 
( ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green" title="E50">A403</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E22:1_1_1_22_1_3"/><i><font color="Green" title="E83">A529</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">j1'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1_22_1_3"><i><font color="Green" title="E78">A524</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E23:1_1_1_22_1_3"/><i><font color="Green" title="E84">A530</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c22">i2'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">j2'</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E19:1_1_1_22_1_3"><i><font color="Green" title="E80">A526</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>assume </b><a NAME="E24:1_1_1_22_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/><a NAME="E25:1_1_1_22_1_3"/><b>then </b><i><font color="Green" title="E85">A531</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c14">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E12:1_1_1_22_1_3"><i><font color="Green" title="E74">A520</font></i></a>, <a class="txt" href="jordan13.html#E21:1_1_1_22_1_3"><i><font color="Green" title="E82">A528</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_1"/><i><font color="Green" title="E86">A532</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E87">A533</font></i>: 
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_1_1_1"/>
( <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> or <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_1"><i><font color="Green" title="E86">A532</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_1"><i><font color="Green" title="E87">A533</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_1"><i><font color="Green" title="E86">A532</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_1"><i><font color="Green" title="E87">A533</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_1_1_1_3"/>
<font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_1"><i><font color="Green" title="E86">A532</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_1"><i><font color="Green" title="E87">A533</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_2"/><i><font color="Green" title="E86">A534</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_2"><i><font color="Green" title="E86">A534</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_2_1_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_2_1_1_2"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_2"><i><font color="Green" title="E86">A534</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_2_1_1_3"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_2"><i><font color="Green" title="E86">A534</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_2_1_1_4"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_2"><i><font color="Green" title="E86">A534</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_3"/><i><font color="Green" title="E86">A535</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_3_1_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_3"><i><font color="Green" title="E86">A535</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_3_1_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_3"><i><font color="Green" title="E86">A535</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_3_1_1_2"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_3"><i><font color="Green" title="E86">A535</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_3_1_1_3"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_3"><i><font color="Green" title="E86">A535</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_3_1_1_4"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_3_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_3"><i><font color="Green" title="E86">A535</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_4"/><i><font color="Green" title="E86">A536</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_4"/><b>then </b><i><font color="Green" title="E87">A537</font></i>: 
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_4_1_1"/>
( <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> or <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_4"><i><font color="Green" title="E86">A536</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_4_1_1_1"/>
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_4"><i><font color="Green" title="E86">A536</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_4"><i><font color="Green" title="E87">A537</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_4_1_1_2"/>
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_4"><i><font color="Green" title="E86">A536</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_4"><i><font color="Green" title="E87">A537</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_4_1_1_3"/>
<font color="Maroon" title="c17">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_4_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_4"><i><font color="Green" title="E86">A536</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_4"><i><font color="Green" title="E87">A537</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_4"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_5"/><i><font color="Green" title="E86">A538</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_5_1_1"/>
( ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_5"><i><font color="Green" title="E86">A538</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_5_1_1_1"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_5_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_5_1_1_2"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_5_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_5"><i><font color="Green" title="E86">A538</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_5_1_1_3"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_5_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_5"><i><font color="Green" title="E86">A538</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_5_1_1_4"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_5_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_5"><i><font color="Green" title="E86">A538</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_5"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_6"/><i><font color="Green" title="E86">A539</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_6"/><b>then </b><i><font color="Green" title="E87">A540</font></i>: 
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_6_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> or <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_6"><i><font color="Green" title="E86">A539</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_6_1_1_1"/>
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_6_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_6"><i><font color="Green" title="E87">A540</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_6_1_1_2"/>
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_6_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_6"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_6"><i><font color="Green" title="E87">A540</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_6_1_1_3"/>
<font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_6_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_6"><i><font color="Green" title="E86">A539</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_6"><i><font color="Green" title="E87">A540</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_6"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_7"/><i><font color="Green" title="E86">A541</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_7"/><b>then </b><i><font color="Green" title="E87">A542</font></i>: 
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_7_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> or <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_7"><i><font color="Green" title="E86">A541</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_7_1_1_1"/>
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_7_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_7"><i><font color="Green" title="E86">A541</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_7"><i><font color="Green" title="E87">A542</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_7_1_1_2"/>
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_7_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_7"><i><font color="Green" title="E86">A541</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_7"><i><font color="Green" title="E87">A542</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_7_1_1_3"/>
<font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_7_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_7"><i><font color="Green" title="E86">A541</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_7"><i><font color="Green" title="E87">A542</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_7"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_8"/><i><font color="Green" title="E86">A543</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_8_1_1"/>
( ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_8"><i><font color="Green" title="E86">A543</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_8_1_1_1"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_8_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_8"><i><font color="Green" title="E86">A543</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_8_1_1_2"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_8_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_8"><i><font color="Green" title="E86">A543</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_8_1_1_3"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_8_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_8"><i><font color="Green" title="E86">A543</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_8_1_1_4"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_8_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_8"><i><font color="Green" title="E86">A543</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_8"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_9"/><i><font color="Green" title="E86">A544</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_9_1_1"/>
( ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_9"><i><font color="Green" title="E86">A544</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_9_1_1_1"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_9_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_9"><i><font color="Green" title="E86">A544</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_9_1_1_2"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_9_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_9"><i><font color="Green" title="E86">A544</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_9_1_1_3"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_9_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_9"><i><font color="Green" title="E86">A544</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_9_1_1_4"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_9_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_9"><i><font color="Green" title="E86">A544</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_9"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_10"/><i><font color="Green" title="E86">A545</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_10"/><b>then </b><i><font color="Green" title="E87">A546</font></i>: 
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_10_1_1"/>
( <font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> or <font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_10"><i><font color="Green" title="E86">A545</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_10_1_1_1"/>
<font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_10_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_10"><i><font color="Green" title="E86">A545</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_10"><i><font color="Green" title="E87">A546</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_10_1_1_2"/>
<font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_10_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_10"><i><font color="Green" title="E86">A545</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_10"><i><font color="Green" title="E87">A546</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_10_1_1_3"/>
<font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_10_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_10"><i><font color="Green" title="E86">A545</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_10"><i><font color="Green" title="E87">A546</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_10"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_11"/><i><font color="Green" title="E86">A547</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_11"/><b>then </b><i><font color="Green" title="E87">A548</font></i>: 
<font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_11_1_1"/>
( <font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> or <font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_11"><i><font color="Green" title="E86">A547</font></i></a>, <a class="ref" href="goboard7.html#T25" title="GOBOARD7:th.25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_11_1_1_1"/>
<font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_11_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_11"><i><font color="Green" title="E86">A547</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_11"><i><font color="Green" title="E87">A548</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_11_1_1_2"/>
<font color="Maroon" title="c18">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_11_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_11"><i><font color="Green" title="E86">A547</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_11"><i><font color="Green" title="E87">A548</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_11_1_1_3"/>
<font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_11_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_11"><i><font color="Green" title="E86">A547</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_11"><i><font color="Green" title="E87">A548</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_11"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_12"/><i><font color="Green" title="E86">A549</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_12_1_1"/>
( ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) or ( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_12"><i><font color="Green" title="E86">A549</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_12_1_1_1"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_12_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_12"><i><font color="Green" title="E86">A549</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_12_1_1_2"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_12_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_12"><i><font color="Green" title="E86">A549</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_12_1_1_3"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_12_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_12"><i><font color="Green" title="E86">A549</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_12_1_1_4"/>
( <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j1</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_12_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_12"><i><font color="Green" title="E86">A549</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_12"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_13"/><i><font color="Green" title="E86">A550</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_13"/><b>then </b><i><font color="Green" title="E87">A551</font></i>: 
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_13_1_1"/>
( <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> or <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_13"><i><font color="Green" title="E86">A550</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_13_1_1_1"/>
<font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_13_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_13"><i><font color="Green" title="E86">A550</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_13"><i><font color="Green" title="E87">A551</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_13_1_1_2"/>
<font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_13_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_13"><i><font color="Green" title="E86">A550</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_13"><i><font color="Green" title="E87">A551</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_13_1_1_3"/>
<font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_13_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_13"><i><font color="Green" title="E86">A550</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_13"><i><font color="Green" title="E87">A551</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_13"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_14"/><i><font color="Green" title="E86">A552</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_14_1_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_14"><i><font color="Green" title="E86">A552</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_14_1_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_14_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_14"><i><font color="Green" title="E86">A552</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_14_1_1_2"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_14_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_14"><i><font color="Green" title="E86">A552</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_14_1_1_3"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_14_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_14"><i><font color="Green" title="E86">A552</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_14_1_1_4"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_14_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_14"><i><font color="Green" title="E86">A552</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_14"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_15"/><i><font color="Green" title="E86">A553</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_15_1_1"/>
( ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) or ( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_15"><i><font color="Green" title="E86">A553</font></i></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_15_1_1_1"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_15_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_15"><i><font color="Green" title="E86">A553</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_15_1_1_2"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_15_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_15"><i><font color="Green" title="E86">A553</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_15_1_1_3"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_15_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_15"><i><font color="Green" title="E86">A553</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_15_1_1_4"/>
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">j1'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_15_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green" title="E59">A459</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green" title="E68">A514</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_15"><i><font color="Green" title="E86">A553</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_15"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_16"/><i><font color="Green" title="E86">A554</font></i>: 
( <font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i2</font> &amp; <font color="Maroon" title="c17">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c20">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">i2'</font> &amp; <font color="Maroon" title="c21">j1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_16"/><b>then </b><i><font color="Green" title="E87">A555</font></i>: 
<font color="Maroon" title="c16">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i1'</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_16_1_1"/>
( <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> or <font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green" title="E66">A512</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green" title="E75">A521</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green" title="E76">A522</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green" title="E77">A523</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green" title="E83">A529</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green" title="E84">A530</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green" title="E85">A531</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_16"><i><font color="Green" title="E86">A554</font></i></a>, <a class="ref" href="goboard7.html#T24" title="GOBOARD7:th.24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_16_1_1_1"/>
<font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_16_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green" title="E53">A428</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green" title="E60">A460</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green" title="E67">A513</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_16"><i><font color="Green" title="E86">A554</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_16"><i><font color="Green" title="E87">A555</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_16_1_1_2"/>
<font color="Maroon" title="c19">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_16_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green" title="E52">A405</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green" title="E57">A457</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green" title="E61">A461</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green" title="E62">A462</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green" title="E65">A511</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green" title="E69">A515</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green" title="E73">A519</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green" title="E79">A525</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_16"><i><font color="Green" title="E86">A554</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_16"><i><font color="Green" title="E87">A555</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_16_1_1_3"/>
<font color="Maroon" title="c19">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">j2'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_16_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green" title="E54">A436</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green" title="E63">A463</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green" title="E64">A510</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green" title="E71">A517</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green" title="E81">A527</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_16"><i><font color="Green" title="E86">A554</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_16"><i><font color="Green" title="E87">A555</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_16"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E27:1_1_1_22_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E74:1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green" title="E47">A400</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E75:1_1_1"/><b>then </b><i><font color="Green" title="E57">A556</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E76:1_1_1"/><b>then </b><i><font color="Green" title="E58">A557</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 
;<br/>
<a NAME="E77:1_1_1"/>
not <font color="Maroon" title="c13">g</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_23"><b>proof </b></a><div class="add">

<b>take </b>
1
; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a><br/>

<b>take </b>
2
;<br/>

<b>thus </b><a NAME="E1:1_1_1_23"/><i><font color="Green" title="E59">A558</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>

<b>thus </b><a NAME="E2:1_1_1_23"/><i><font color="Green" title="E60">A559</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E75:1_1_1"><i><font color="Green" title="E57">A556</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>

<a NAME="E3:1_1_1_23"/><b>then </b>
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E68:1_1_1"><i><font color="Green" title="E51">A404</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_23"><i><font color="Green" title="E59">A558</font></i></a>, <a class="ref" href="goboard7.html#T31" title="GOBOARD7:th.31">GOBOARD7:31</a></i>;<br/>
<a NAME="E4:1_1_1_23"/><b>then </b>
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_23"><i><font color="Green" title="E59">A558</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_1_23"/>
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_23"><i><font color="Green" title="E60">A559</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">g</font> as  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a> <b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green" title="E38">A391</font></i></a>, <a class="txt" href="jordan13.html#E64:1_1_1"><i><font color="Green" title="E48">A401</font></i></a>, <a class="txt" href="jordan13.html#E66:1_1_1"><i><font color="Green" title="E49">A402</font></i></a>, <a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green" title="E50">A403</font></i></a>, <a class="txt" href="jordan13.html#E73:1_1_1"><i><font color="Green" title="E56">A456</font></i></a>, <a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a>, <a class="ref" href="jordan8.html#T7" title="JORDAN8:th.7">JORDAN8:7</a></i>;<br/>
<a NAME="E79:1_1_1"/><i><font color="Green" title="E59">A560</font></i>: 
for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_24"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">j</font>, <font color="Maroon" title="c16">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_24"/><i><font color="Green" title="E60">A561</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font>
 <b>and </b><br/><a NAME="E2:1_1_1_24"/><i><font color="Green" title="E61">A562</font></i>: 
<font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font>
 ;<br/>

<a NAME="E3:1_1_1_24"/><i><font color="Green" title="E62">A563</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c15">j</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green" title="E28">A321</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_24"><i><font color="Green" title="E61">A562</font></i></a></i>;<br/>
<a NAME="E4:1_1_1_24"/>
<font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_24"><i><font color="Green" title="E61">A562</font></i></a></i>;<br/>
<a NAME="E5:1_1_1_24"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E6:1_1_1_24"/><b>then </b>
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c15">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_24"><i><font color="Green" title="E60">A561</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_1_24"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_24"><i><font color="Green" title="E62">A563</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c15">Lg'</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E81:1_1_1"/><i><font color="Green" title="E60">A564</font></i>: 
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Lg'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_25"/><i><font color="Green" title="E61">A565</font></i>: 
<font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:1_1_1_25"/><i><font color="Green" title="E62">A566</font></i>: 
not <font color="Maroon" title="c16">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Lg'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c17">c</font> = <font color="Maroon" title="c16">c</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_25"><i><font color="Green" title="E61">A565</font></i></a></i>;<br/>
<a NAME="E4:1_1_1_25"/>
<font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_25"><i><font color="Green" title="E62">A566</font></i></a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:1_1_1_25"/><i><font color="Green" title="E63">A567</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> )
 <b>and </b><br/><a NAME="E7:1_1_1_25"/><i><font color="Green" title="E64">A568</font></i>: 
<font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T14" title="SPPOL_2:th.14">SPPOL_2:14</a></i>;<br/>
<a NAME="E8:1_1_1_25"/>
( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> &amp; <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_25"><i><font color="Green" title="E63">A567</font></i></a>, <a class="ref" href="goboard2.html#T3" title="GOBOARD2:th.3">GOBOARD2:3</a></i>;<br/>
<a NAME="E9:1_1_1_25"/><b>then </b><i><font color="Green" title="E65">A569</font></i>: 
( <font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>;<br/>
<a NAME="E10:1_1_1_25"/><i><font color="Green" title="E66">A570</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_25"><i><font color="Green" title="E63">A567</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E11:1_1_1_25"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E12:1_1_1_25"/><b>then </b><i><font color="Green" title="E67">A571</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_25"><i><font color="Green" title="E63">A567</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E13:1_1_1_25"/><b>then </b>
<font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green" title="E47">A400</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_25"><i><font color="Green" title="E64">A568</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_25"><i><font color="Green" title="E65">A569</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_25"><i><font color="Green" title="E66">A570</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E14:1_1_1_25"/><b>then </b>
<font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_25"><i><font color="Green" title="E66">A570</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_25"><i><font color="Green" title="E67">A571</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green" title="E47">A400</font></i></a>, <a class="ref" href="gobrd13.html#T30" title="GOBRD13:th.30">GOBRD13:30</a></i>;<br/>
<a NAME="E15:1_1_1_25"/><b>then </b>
<font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E16:1_1_1_25"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_25"><i><font color="Green" title="E61">A565</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E17:1_1_1_25"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_25"><i><font color="Green" title="E66">A570</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_25"><i><font color="Green" title="E67">A571</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green" title="E47">A400</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E82:1_1_1"/><i><font color="Green" title="E61">A572</font></i>: 
 <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></i>;<br/>
<a NAME="E83:1_1_1"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="sppol_1.html#T49" title="SPPOL_1:th.49">SPPOL_1:49</a></i>;<br/>
<a NAME="E84:1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>
<a NAME="E85:1_1_1"/><b>then </b><i><font color="Green" title="E62">A573</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/>
<a NAME="E86:1_1_1"/><i><font color="Green" title="E63">A574</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c11">f</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T75" title="JORDAN3:th.75">JORDAN3:75</a></i>;<br/>
<a NAME="E87:1_1_1"/><i><font color="Green" title="E64">A575</font></i>: 
(  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c14">g</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp;  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c14">g</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a>, <a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></i>;<br/>
<a NAME="E88:1_1_1"/><i><font color="Green" title="E65">A576</font></i>: 
for <font color="Olive" title="b1">h</font> being non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>  st  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b1">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c11">f</font> holds <br/>for <font color="Olive" title="b2">Comp</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">Comp</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  holds <br/>for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> &amp; <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Comp</font> &amp; not <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Olive" title="b1">h</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">Comp</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_26"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">h</font> be  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>;<br/>

<b>assume </b><a NAME="E1:1_1_1_26"/><i><font color="Green" title="E66">A577</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c11">f</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c17">Comp</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E2:1_1_1_26"/><i><font color="Green" title="E67">A578</font></i>: 
<font color="Maroon" title="c17">Comp</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c18">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:1_1_1_26"/><i><font color="Green" title="E68">A579</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n</font> &amp; <font color="Maroon" title="c18">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> )
 <b>and </b><br/><a NAME="E4:1_1_1_26"/><i><font color="Green" title="E69">A580</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Comp</font>
 <b>and </b><br/><a NAME="E5:1_1_1_26"/><i><font color="Green" title="E70">A581</font></i>: 
not <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">h</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c19">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">h</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">h</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E7:1_1_1_26"/><i><font color="Green" title="E71">A582</font></i>: 
<font color="Maroon" title="c20">rc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E8:1_1_1_26"/><i><font color="Green" title="E72">A583</font></i>: 
<font color="Maroon" title="c20">rc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T32" title="SUBSET_1:th.32">SUBSET_1:32</a></i>;<br/>
<a NAME="E9:1_1_1_26"/><i><font color="Green" title="E73">A584</font></i>: 
<font color="Maroon" title="c20">rc</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_26_1"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_26_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26"><i><font color="Green" title="E68">A579</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1_26_1"/><i><font color="Green" title="E74">A585</font></i>: 
( <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">p</font> = <font color="Maroon" title="c21">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_26_1"><i><font color="Green" title="E74">A585</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_26_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c22">p</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_26_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_26_1_1_1"/>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">h</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c24">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_26_1_1_1"/><i><font color="Green" title="E75">A586</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">j</font>
 <b>and </b><br/><a NAME="E4:1_1_1_26_1_1_1"/><i><font color="Green" title="E76">A587</font></i>: 
<font color="Maroon" title="c24">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font>
 <b>and </b><br/><a NAME="E5:1_1_1_26_1_1_1"/><i><font color="Green" title="E77">A588</font></i>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c24">j</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_26"><i><font color="Green" title="E66">A577</font></i></a>, <a class="ref" href="sppol_2.html#T13" title="SPPOL_2:th.13">SPPOL_2:13</a></i>;<br/><a NAME="E6:1_1_1_26_1_1_1"/>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c24">j</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c24">j</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26_1_1_1"><i><font color="Green" title="E75">A586</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_26_1_1_1"><i><font color="Green" title="E76">A587</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_26_1_1_1"><i><font color="Green" title="E77">A588</font></i></a>, <a class="ref" href="gobrd13.html#T30" title="GOBRD13:th.30">GOBRD13:30</a></i>;<br/><a NAME="E7:1_1_1_26_1_1_1"/><b>then </b><i><font color="Green" title="E78">A589</font></i>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c24">j</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:1_1_1_26_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c24">j</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26_1_1_1"><i><font color="Green" title="E75">A586</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_26_1_1_1"><i><font color="Green" title="E76">A587</font></i></a></i>;<br/><b>hence </b><a NAME="E9:1_1_1_26_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_26_1"><i><font color="Green" title="E74">A585</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_26_1_1_1"><i><font color="Green" title="E78">A589</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_26_1_1"/>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">rc</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_26_1"><i><font color="Green" title="E74">A585</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_26_1_1"/>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_26_1"><i><font color="Green" title="E74">A585</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:1_1_1_26_1"/>
<font color="Maroon" title="c20">rc</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:1_1_1_26"/><i><font color="Green" title="E74">A590</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26"><i><font color="Green" title="E68">A579</font></i></a>, <a class="ref" href="jordan9.html#T12" title="JORDAN9:th.12">JORDAN9:12</a></i>;<br/>
<a NAME="E11:1_1_1_26"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c11">f</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26"><i><font color="Green" title="E68">A579</font></i></a>, <a class="ref" href="jordan9.html#T17" title="JORDAN9:th.17">JORDAN9:17</a></i>;<br/>
<a NAME="E12:1_1_1_26"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">h</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_26"><i><font color="Green" title="E66">A577</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E13:1_1_1_26"/><b>then </b><i><font color="Green" title="E75">A591</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>
<a NAME="E14:1_1_1_26"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<a NAME="E15:1_1_1_26"/><b>then </b><i><font color="Green" title="E76">A592</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">rc</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_26"><i><font color="Green" title="E72">A583</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_26"><i><font color="Green" title="E75">A591</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E16:1_1_1_26"/>
<font color="Maroon" title="c20">rc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26"><i><font color="Green" title="E68">A579</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_26"><i><font color="Green" title="E71">A582</font></i></a>, <a class="ref" href="jordan9.html#T13" title="JORDAN9:th.13">JORDAN9:13</a></i>;<br/>
<a NAME="E17:1_1_1_26"/><b>then </b><i><font color="Green" title="E77">A593</font></i>: 
<font color="Maroon" title="c20">rc</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_26"><i><font color="Green" title="E74">A590</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_26"><i><font color="Green" title="E76">A592</font></i></a>, <a class="ref" href="connsp_1.html#T19" title="CONNSP_1:th.19">CONNSP_1:19</a></i>;<br/>
<a NAME="E18:1_1_1_26"/>
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26"><i><font color="Green" title="E68">A579</font></i></a>, <a class="ref" href="jordan9.html#T10" title="JORDAN9:th.10">JORDAN9:10</a></i>;<br/>
<a NAME="E19:1_1_1_26"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">rc</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_26"><i><font color="Green" title="E70">A581</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E20:1_1_1_26"/><b>then </b><i><font color="Green" title="E78">A594</font></i>: 
<font color="Maroon" title="c20">rc</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">Comp</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_26"><i><font color="Green" title="E69">A580</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E21:1_1_1_26"/>
<font color="Maroon" title="c20">rc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_26"><i><font color="Green" title="E72">A583</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E22:1_1_1_26"/><b>then </b>
<font color="Maroon" title="c20">rc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">Comp</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_26"><i><font color="Green" title="E67">A578</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_26"><i><font color="Green" title="E77">A593</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_26"><i><font color="Green" title="E78">A594</font></i></a>, <a class="ref" href="goboard9.html#T6" title="GOBOARD9:th.6">GOBOARD9:6</a></i>;<br/>
<b>hence </b><a NAME="E23:1_1_1_26"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">Comp</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_26"><i><font color="Green" title="E73">A584</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E89:1_1_1"/><i><font color="Green" title="E66">A595</font></i>: 
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c14">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_27"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_27"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1_27"/><i><font color="Green" title="E67">A596</font></i>: 
( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <font color="Maroon" title="c16">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">p</font> = <font color="Maroon" title="c16">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_27"><i><font color="Green" title="E67">A596</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_27_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c18">p</font> = <font color="Maroon" title="c17">p</font>;<br/><b>thus </b><a NAME="E1:1_1_1_27_1"/>
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_27"><i><font color="Green" title="E67">A596</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">u</font> = <font color="Maroon" title="c18">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">r</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:1_1_1_27_1"/><i><font color="Green" title="E68">A597</font></i>: 
<font color="Maroon" title="c20">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:1_1_1_27_1"/><i><font color="Green" title="E69">A598</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c20">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan13.html#E81:1_1_1"><i><font color="Green" title="E60">A564</font></i></a>, <a class="txt" href="jordan13.html#E85:1_1_1"><i><font color="Green" title="E62">A573</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_27"><i><font color="Green" title="E67">A596</font></i></a>, <a class="ref" href="goboard6.html#T8" title="GOBOARD6:th.8">GOBOARD6:8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">r</font> = <font color="Maroon" title="c20">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E7:1_1_1_27_1"/><i><font color="Green" title="E70">A599</font></i>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c21">r</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_27_1"><i><font color="Green" title="E68">A597</font></i></a>, <a class="ref" href="tbsp_1.html#T16" title="TBSP_1:th.16">TBSP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">B</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c21">r</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan13.html#E82:1_1_1"><i><font color="Green" title="E61">A572</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_27_1"><i><font color="Green" title="E68">A597</font></i></a>, <a class="ref" href="tbsp_1.html#T16" title="TBSP_1:th.16">TBSP_1:16</a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/><a NAME="E9:1_1_1_27_1"/><i><font color="Green" title="E71">A600</font></i>: 
<font color="Maroon" title="c22">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="sprect_3.html#T17" title="SPRECT_3:th.17">SPRECT_3:17</a></i>;<br/><a NAME="E10:1_1_1_27_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">g</font>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c14">g</font>,1
 <b>by </b><i><a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green" title="E50">A403</font></i></a>, <a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green" title="E58">A557</font></i></a>, <a class="ref" href="gobrd13.html#T34" title="GOBRD13:th.34">GOBRD13:34</a></i>;<br/><a NAME="E11:1_1_1_27_1"/><b>then </b><i><font color="Green" title="E72">A601</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">g</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c14">g</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T48" title="TOPS_1:th.48">TOPS_1:48</a></i>;<br/><a NAME="E12:1_1_1_27_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c14">g</font>,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green" title="E58">A557</font></i></a>, <a class="ref" href="goboard9.html#T24" title="GOBOARD9:th.24">GOBOARD9:24</a></i>;<br/><a NAME="E13:1_1_1_27_1"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c14">g</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_27_1"><i><font color="Green" title="E72">A601</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:1_1_1_27_1"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green" title="E58">A557</font></i></a>, <a class="ref" href="gobrd13.html#T33" title="GOBRD13:th.33">GOBRD13:33</a></i>;<br/><a NAME="E15:1_1_1_27_1"/><b>then </b><i><font color="Green" title="E73">A602</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c9">m</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E16:1_1_1_27_1"/><i><font color="Green" title="E74">A603</font></i>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c9">m</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a>, <a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green" title="E55">A442</font></i></a></i>;<br/><a NAME="E17:1_1_1_27_1"/>
<font color="Maroon" title="c22">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="goboard6.html#T6" title="GOBOARD6:th.6">GOBOARD6:6</a></i>;<br/><a NAME="E18:1_1_1_27_1"/><b>then </b><i><font color="Green" title="E75">A604</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c9">m</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_27"><i><font color="Green" title="E67">A596</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_27_1"><i><font color="Green" title="E70">A599</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_27_1"><i><font color="Green" title="E74">A603</font></i></a>, <a class="ref" href="tops_1.html#T39" title="TOPS_1:th.39">TOPS_1:39</a></i>;<br/><a NAME="E19:1_1_1_27_1"/><i><font color="Green" title="E76">A605</font></i>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_27_1"><i><font color="Green" title="E68">A597</font></i></a>, <a class="ref" href="tbsp_1.html#T16" title="TBSP_1:th.16">TBSP_1:16</a></i>;<br/><a NAME="E20:1_1_1_27_1"/>
<font color="Maroon" title="c22">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E87:1_1_1"><i><font color="Green" title="E64">A575</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_27_1"><i><font color="Green" title="E69">A598</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_27_1"><i><font color="Green" title="E71">A600</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_27_1"><i><font color="Green" title="E73">A602</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_27_1"><i><font color="Green" title="E75">A604</font></i></a>, <a class="ref" href="goboard9.html#T6" title="GOBOARD9:th.6">GOBOARD9:6</a></i>;<br/><b>hence </b><a NAME="E21:1_1_1_27_1"/>
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E19:1_1_1_27_1"><i><font color="Green" title="E76">A605</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:1_1_1_27"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E90:1_1_1"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_28"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:1_1_1_28"/><b>then </b><i><font color="Green" title="E67">A606</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">m</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E3:1_1_1_28"/><i><font color="Green" title="E68">A607</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 holds <br/> not <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_28_1"/><i><font color="Green" title="E69">A608</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 <b>and </b><br/><a NAME="E2:1_1_1_28_1"/><i><font color="Green" title="E70">A609</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 ;<br/>

<a NAME="E3:1_1_1_28_1"/><i><font color="Green" title="E71">A610</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_1"><i><font color="Green" title="E70">A609</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">p</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font>;<br/>
<b>assume </b><a NAME="E4:1_1_1_28_1"/>
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:1_1_1_28_1"/><i><font color="Green" title="E72">A611</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font>
 <b>and </b><br/><a NAME="E7:1_1_1_28_1"/><i><font color="Green" title="E73">A612</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font>
 <b>and </b><br/><a NAME="E8:1_1_1_28_1"/><i><font color="Green" title="E74">A613</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c11">f</font>,<font color="Maroon" title="c18">j</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="ref" href="jordan9.html#T9" title="JORDAN9:th.9">JORDAN9:9</a></i>;<br/>
<a NAME="E9:1_1_1_28_1"/><i><font color="Green" title="E75">A614</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_1"><i><font color="Green" title="E70">A609</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E10:1_1_1_28_1"/><b>then </b><i><font color="Green" title="E76">A615</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">j</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_28_1"><i><font color="Green" title="E72">A611</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:1_1_1_28_1"/><i><font color="Green" title="E77">A616</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E12:1_1_1_28_1"/><b>then </b><i><font color="Green" title="E78">A617</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">j</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green" title="E67">A606</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_1"><i><font color="Green" title="E72">A611</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:1_1_1_28_1"/><i><font color="Green" title="E79">A618</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_1"><i><font color="Green" title="E73">A612</font></i></a></i>;<br/>
<a NAME="E14:1_1_1_28_1"/><i><font color="Green" title="E80">A619</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_1"><i><font color="Green" title="E73">A612</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E15:1_1_1_28_1"/><i><font color="Green" title="E81">A620</font></i>: 
( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E16:1_1_1_28_1"/><i><font color="Green" title="E82">A621</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green" title="E69">A608</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1"><i><font color="Green" title="E71">A610</font></i></a>, <a class="ref" href="jordan9.html#T8" title="JORDAN9:th.8">JORDAN9:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_28_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">j</font> or <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_1"><i><font color="Green" title="E73">A612</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_1"><i><font color="Green" title="E74">A613</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_28_1"><i><font color="Green" title="E78">A617</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_1"><i><font color="Green" title="E81">A620</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_1"><i><font color="Green" title="E82">A621</font></i></a>, <a class="ref" href="jordan9.html#T25" title="JORDAN9:th.25">JORDAN9:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_1_1_1"/><i><font color="Green" title="E83">A622</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">j</font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_1_1_1"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_28_1"><i><font color="Green" title="E76">A615</font></i></a></i>;<br/><a NAME="E3:1_1_1_28_1_1_1"/><b>then </b><i><font color="Green" title="E84">A623</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green" title="E69">A608</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:1_1_1_28_1_1_1"/><i><font color="Green" title="E85">A624</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_1"><i><font color="Green" title="E72">A611</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_1"><i><font color="Green" title="E75">A614</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green" title="E69">A608</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_28_1"><i><font color="Green" title="E76">A615</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green" title="E69">A608</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1"><i><font color="Green" title="E71">A610</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c18">j</font>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_28_1"><i><font color="Green" title="E78">A617</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_1"><i><font color="Green" title="E80">A619</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_1"><i><font color="Green" title="E83">A622</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:1_1_1_28_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green" title="E35">A388</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_28_1"><i><font color="Green" title="E78">A617</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_1"><i><font color="Green" title="E80">A619</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1_1_1"><i><font color="Green" title="E84">A623</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_1_1_1"><i><font color="Green" title="E85">A624</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_1_1_2"/><i><font color="Green" title="E83">A625</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_28_1_1_2_1_1"/>
( <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> or <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">k</font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E13:1_1_1_28_1"><i><font color="Green" title="E79">A618</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_1_1_2_1_1_1"/><i><font color="Green" title="E84">A626</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_1_1_2_1_1_1"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_1"><i><font color="Green" title="E75">A614</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_28_1"><i><font color="Green" title="E77">A616</font></i></a></i>;<br/><a NAME="E3:1_1_1_28_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E85">A627</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green" title="E69">A608</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:1_1_1_28_1_1_2_1_1_1"/><i><font color="Green" title="E86">A628</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_1"><i><font color="Green" title="E75">A614</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_28_1"><i><font color="Green" title="E77">A616</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green" title="E69">A608</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_1"><i><font color="Green" title="E75">A614</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_28_1"><i><font color="Green" title="E77">A616</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green" title="E69">A608</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1"><i><font color="Green" title="E71">A610</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">m</font>
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green" title="E38">A391</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_2"><i><font color="Green" title="E83">A625</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_2_1_1_1"><i><font color="Green" title="E84">A626</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:1_1_1_28_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green" title="E35">A388</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green" title="E42">A395</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1_1_2_1_1_1"><i><font color="Green" title="E85">A627</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_1_1_2_1_1_1"><i><font color="Green" title="E86">A628</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_1_1_2_1_1_2"/><i><font color="Green" title="E84">A629</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">k</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c19">l</font> = <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><a NAME="E2:1_1_1_28_1_1_2_1_1_2"/><i><font color="Green" title="E85">A630</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:1_1_1_28_1_1_2_1_1_2"/><i><font color="Green" title="E86">A631</font></i>: 
( <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_28_1"><i><font color="Green" title="E76">A615</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E4:1_1_1_28_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E87">A632</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:1_1_1_28_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/><a NAME="E6:1_1_1_28_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E88">A633</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green" title="E69">A608</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:1_1_1_28_1_1_2_1_1_2"/><i><font color="Green" title="E89">A634</font></i>: 
<font color="Maroon" title="c16">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1_1_2_1_1_2"><i><font color="Green" title="E86">A631</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green" title="E69">A608</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_1_1_2_1_1_2"><i><font color="Green" title="E87">A632</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c16">n</font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green" title="E69">A608</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1"><i><font color="Green" title="E71">A610</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_2"><i><font color="Green" title="E83">A625</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_2_1_1_2"><i><font color="Green" title="E84">A629</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_1_1_2_1_1_2"><i><font color="Green" title="E85">A630</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>
;<br/><b>hence </b><a NAME="E9:1_1_1_28_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green" title="E35">A388</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_2_1_1_2"><i><font color="Green" title="E84">A629</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_1_1_2_1_1_2"><i><font color="Green" title="E85">A630</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_1_1_2_1_1_2"><i><font color="Green" title="E88">A633</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_1_1_2_1_1_2"><i><font color="Green" title="E89">A634</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_28_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:1_1_1_28"/><i><font color="Green" title="E69">A635</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_1_28_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T56" title="BINARITH:th.56">BINARITH:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T55" title="BINARITH:th.55">BINARITH:55</a></i>
<br/>.= 
<font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1

;<br/><br/>


</div><b>end;</b></div>
<a NAME="E5:1_1_1_28"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c16">rg</font> =  <a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font>;<br/>
<a NAME="E1:1_1_1_28_3"/><i><font color="Green" title="E70">A636</font></i>: 
 <a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green" title="E50">A403</font></i></a>, <a class="ref" href="jordan9.html#T7" title="JORDAN9:th.7">JORDAN9:7</a></i>;<br/>
<a NAME="E2:1_1_1_28_3"/><i><font color="Green" title="E71">A637</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E75:1_1_1"><i><font color="Green" title="E57">A556</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E3:1_1_1_28_3"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/>
<a NAME="E4:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E72">A638</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E5:1_1_1_28_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3"><i><font color="Green" title="E71">A637</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E73">A639</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">p</font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1;<br/>
<b>set </b><font color="Maroon" title="c18">q</font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2;<br/>
<b>set </b><font color="Maroon" title="c19">a</font> = <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c20">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>;<br/>
<a NAME="E7:1_1_1_28_3"/><i><font color="Green" title="E74">A640</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">m</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green" title="E38">A391</font></i></a>, <a class="txt" href="jordan13.html#E66:1_1_1"><i><font color="Green" title="E49">A402</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>;<br/>
<a NAME="E8:1_1_1_28_3"/><i><font color="Green" title="E75">A641</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E9:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E76">A642</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green" title="E67">A606</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E10:1_1_1_28_3"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E75:1_1_1"><i><font color="Green" title="E57">A556</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/>
<a NAME="E11:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E77">A643</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E12:1_1_1_28_3"/><i><font color="Green" title="E78">A644</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E75:1_1_1"><i><font color="Green" title="E57">A556</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E13:1_1_1_28_3"/><i><font color="Green" title="E79">A645</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green" title="E67">A606</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<i><font color="Green" title="E80">A646</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="ref" href="binarith.html#T56" title="BINARITH:th.56">BINARITH:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_28_3"><i><font color="Green" title="E77">A643</font></i></a>, <a class="ref" href="binarith.html#T55" title="BINARITH:th.55">BINARITH:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E12:1_1_1_28_3"><i><font color="Green" title="E78">A644</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green" title="E47">A400</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_28_3"><i><font color="Green" title="E79">A645</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

;<br/>
<a NAME="E15:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E81">A647</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E16:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E82">A648</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">k</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E17:1_1_1_28_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E18:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E83">A649</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_28_3"><i><font color="Green" title="E77">A643</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:1_1_1_28_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/>
<a NAME="E20:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E84">A650</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_28_3"><i><font color="Green" title="E77">A643</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E21:1_1_1_28_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green" title="E58">A557</font></i></a>, <a class="ref" href="sppol_1.html#T5" title="SPPOL_1:th.5">SPPOL_1:5</a></i>;<br/>
<a NAME="E22:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E85">A651</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_28"><i><font color="Green" title="E69">A635</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E86">A652</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 = 
<font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E20:1_1_1_28_3"><i><font color="Green" title="E84">A650</font></i></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E20:1_1_1_28_3"><i><font color="Green" title="E84">A650</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/>
<a NAME="E24:1_1_1_28_3"/><i><font color="Green" title="E87">A653</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E66:1_1_1"><i><font color="Green" title="E49">A402</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>;<br/>
<a NAME="E25:1_1_1_28_3"/><i><font color="Green" title="E88">A654</font></i>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a></i>;<br/>
<a NAME="E26:1_1_1_28_3"/><i><font color="Green" title="E89">A655</font></i>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green" title="E81">A647</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a></i>;<br/>
<a NAME="E27:1_1_1_28_3"/>
not <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_28"><i><font color="Green" title="E68">A607</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a></i>;<br/>
<a NAME="E28:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E90">A656</font></i>: 
not <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/>
<a NAME="E29:1_1_1_28_3"/><i><font color="Green" title="E91">A657</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28"><i><font color="Green" title="E69">A635</font></i></a></i>;<br/>
<a NAME="E30:1_1_1_28_3"/><i><font color="Green" title="E92">A658</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E31:1_1_1_28_3"/><b>then </b>
<font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/>
<a NAME="E32:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E93">A659</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E33:1_1_1_28_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_28_3"><i><font color="Green" title="E77">A643</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E34:1_1_1_28_3"/><b>then </b>
<font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E35:1_1_1_28_3"/><b>then </b><i><font color="Green" title="E94">A660</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">p1</font>, <font color="Maroon" title="c22">p2</font>, <font color="Maroon" title="c23">q1</font>, <font color="Maroon" title="c24">q2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E37:1_1_1_28_3"/><i><font color="Green" title="E95">A661</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">p1</font>,<font color="Maroon" title="c22">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E38:1_1_1_28_3"/><i><font color="Green" title="E96">A662</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">p1</font>,<font color="Maroon" title="c22">p2</font>
 <b>and </b><br/><a NAME="E39:1_1_1_28_3"/><i><font color="Green" title="E97">A663</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">q1</font>,<font color="Maroon" title="c24">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E40:1_1_1_28_3"/><i><font color="Green" title="E98">A664</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c23">q1</font>,<font color="Maroon" title="c24">q2</font>
 <b>and </b><br/><a NAME="E41:1_1_1_28_3"/><i><font color="Green" title="E99">A665</font></i>: 
( ( <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> &amp; <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> ) or ( <font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> &amp; <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> ) or ( <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> ) or ( <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> &amp; <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green" title="E70">A636</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3"><i><font color="Green" title="E71">A637</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/>
<a NAME="E42:1_1_1_28_3"/><i><font color="Green" title="E100">A666</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c21">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">p2</font> &amp; <font color="Maroon" title="c22">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_28_3_3"/>
( ( <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> &amp; <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> ) or ( <font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> &amp; <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> ) or ( <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> ) or ( <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> &amp; <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green" title="E99">A665</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_1"/><i><font color="Green" title="E101">A667</font></i>: 
( <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> &amp; <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c25">a1</font>, <font color="Maroon" title="c26">a2</font>, <font color="Maroon" title="c27">p'1</font>, <font color="Maroon" title="c28">p'2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_28_3_3_1"/><i><font color="Green" title="E102">A668</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">a1</font>,<font color="Maroon" title="c26">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_28_3_3_1"/><i><font color="Green" title="E103">A669</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">a1</font>,<font color="Maroon" title="c26">a2</font>
 <b>and </b><br/><a NAME="E5:1_1_1_28_3_3_1"/><i><font color="Green" title="E104">A670</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">p'1</font>,<font color="Maroon" title="c28">p'2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_28_3_3_1"/><i><font color="Green" title="E105">A671</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">p'1</font>,<font color="Maroon" title="c28">p'2</font>
 <b>and </b><br/><a NAME="E7:1_1_1_28_3_3_1"/><i><font color="Green" title="E106">A672</font></i>: 
( ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_28_3_3_1"/><i><font color="Green" title="E107">A673</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">a1</font> &amp; <font color="Maroon" title="c25">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">a2</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1"><i><font color="Green" title="E102">A668</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_28_3_3_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_28_3_3_1_1_1"/>
( ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_1"><i><font color="Green" title="E106">A672</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_1_1_1_1"/>
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E108">A674</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1"><i><font color="Green" title="E104">A670</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1"><i><font color="Green" title="E105">A671</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green" title="E109">A675</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/><a NAME="E4:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green" title="E110">A676</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a></i>;<br/><a NAME="E5:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green" title="E111">A677</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green" title="E112">A678</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green" title="E28">A321</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a></i>;<br/><a NAME="E7:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green" title="E113">A679</font></i>: 
( <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c26">a2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E108">A674</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E114">A680</font></i>: <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E111">A677</font></i></a></i>
;<br/><a NAME="E9:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green" title="E115">A681</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a></i>;<br/> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c21">p1</font>,<font color="Maroon" title="c22">p2</font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1"><i><font color="Green" title="E101">A667</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1"><i><font color="Green" title="E102">A668</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green" title="E103">A669</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E108">A674</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E109">A675</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E110">A676</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E114">A680</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E115">A681</font></i></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></i>
;<br/><a NAME="E11:1_1_1_28_3_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green" title="E29">A322</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green" title="E67">A606</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green" title="E89">A655</font></i></a></i>;<br/><a NAME="E12:1_1_1_28_3_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E112">A678</font></i></a>, <a class="ref" href="gobrd13.html#T44" title="GOBRD13:th.44">GOBRD13:44</a></i>;<br/><a NAME="E13:1_1_1_28_3_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E116">A682</font></i>: 
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">p2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c22">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1"><i><font color="Green" title="E102">A668</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green" title="E103">A669</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E113">A679</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><a NAME="E14:1_1_1_28_3_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E117">A683</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>set </b><font color="Maroon" title="c29">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>;<br/><a NAME="E15:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green" title="E118">A684</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green" title="E70">A636</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green" title="E72">A638</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green" title="E73">A639</font></i></a>, <a class="ref" href="jordan9.html#T29" title="JORDAN9:th.29">JORDAN9:29</a></i>;<br/><a NAME="E16:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green" title="E119">A685</font></i>: 
( <font color="Maroon" title="c22">p2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1_28_3"><i><font color="Green" title="E100">A666</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E108">A674</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E17:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green" title="E120">A686</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E75:1_1_1"><i><font color="Green" title="E57">A556</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:1_1_1_28_3_3_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E22:1_1_1_28_3"><i><font color="Green" title="E85">A651</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E121">A687</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E120">A686</font></i></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E120">A686</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green" title="E38">A391</font></i></a>, <a class="txt" href="jordan13.html#E64:1_1_1"><i><font color="Green" title="E48">A401</font></i></a>, <a class="txt" href="jordan13.html#E66:1_1_1"><i><font color="Green" title="E49">A402</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E21:1_1_1_28_3_3_1_1_1_1"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c21">p1</font>,<font color="Maroon" title="c26">a2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green" title="E70">A636</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green" title="E72">A638</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green" title="E73">A639</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E116">A682</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E119">A685</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E121">A687</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/><a NAME="E22:1_1_1_28_3_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1_28_3"><i><font color="Green" title="E100">A666</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green" title="E103">A669</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_1"><i><font color="Green" title="E107">A673</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E108">A674</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E117">A683</font></i></a>, <a class="ref" href="jordan9.html#T22" title="JORDAN9:th.22">JORDAN9:22</a></i>;<br/><a NAME="E23:1_1_1_28_3_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E122">A688</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1_28_3"><i><font color="Green" title="E90">A656</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E24:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green" title="E123">A689</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c11">f</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E86:1_1_1"><i><font color="Green" title="E63">A574</font></i></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/><a NAME="E25:1_1_1_28_3_3_1_1_1_1"/>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></i>;<br/><b>hence </b><a NAME="E26:1_1_1_28_3_3_1_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E88:1_1_1"><i><font color="Green" title="E65">A576</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_28_3"><i><font color="Green" title="E90">A656</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E118">A684</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E122">A688</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3_3_1_1_1_1"><i><font color="Green" title="E123">A689</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_1_1_1_2"/><i><font color="Green" title="E108">A690</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E109">A691</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1"><i><font color="Green" title="E104">A670</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1"><i><font color="Green" title="E105">A671</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c23">q1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">a1</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1"><i><font color="Green" title="E101">A667</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><b>then </b> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c25">a1</font>,<font color="Maroon" title="c26">a2</font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1"><i><font color="Green" title="E101">A667</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_2"><i><font color="Green" title="E109">A691</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1"><i><font color="Green" title="E102">A668</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green" title="E103">A669</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1"><i><font color="Green" title="E104">A670</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1"><i><font color="Green" title="E105">A671</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1_1_1_2"><i><font color="Green" title="E108">A690</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>
;<br/><b>hence </b><a NAME="E5:1_1_1_28_3_3_1_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green" title="E81">A647</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green" title="E88">A654</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_1_1_1_3"/>
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1"><i><font color="Green" title="E104">A670</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1"><i><font color="Green" title="E105">A671</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><b>then </b> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c21">p1</font>,<font color="Maroon" title="c22">p2</font>
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1"><i><font color="Green" title="E102">A668</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green" title="E103">A669</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1"><i><font color="Green" title="E101">A667</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1_28_3_3_1_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green" title="E89">A655</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_1_1_1_4"/>
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_4"/><b>then </b><i><font color="Green" title="E108">A692</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1"><i><font color="Green" title="E101">A667</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1"><i><font color="Green" title="E104">A670</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1"><i><font color="Green" title="E105">A671</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green" title="E92">A658</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green" title="E103">A669</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_4"><i><font color="Green" title="E108">A692</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green" title="E82">A648</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1_28_3_3_1_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green" title="E35">A388</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green" title="E82">A648</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E32:1_1_1_28_3"><i><font color="Green" title="E93">A659</font></i></a>, <a class="txt" href="jordan13.html#E35:1_1_1_28_3"><i><font color="Green" title="E94">A660</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_2"/><i><font color="Green" title="E101">A693</font></i>: 
( <font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> &amp; <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c25">a1</font>, <font color="Maroon" title="c26">a2</font>, <font color="Maroon" title="c27">p'1</font>, <font color="Maroon" title="c28">p'2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_28_3_3_2"/><i><font color="Green" title="E102">A694</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">a1</font>,<font color="Maroon" title="c26">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_28_3_3_2"/><i><font color="Green" title="E103">A695</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">a1</font>,<font color="Maroon" title="c26">a2</font>
 <b>and </b><br/><a NAME="E5:1_1_1_28_3_3_2"/><i><font color="Green" title="E104">A696</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">p'1</font>,<font color="Maroon" title="c28">p'2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_28_3_3_2"/><i><font color="Green" title="E105">A697</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">p'1</font>,<font color="Maroon" title="c28">p'2</font>
 <b>and </b><br/><a NAME="E7:1_1_1_28_3_3_2"/><i><font color="Green" title="E106">A698</font></i>: 
( ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_28_3_3_2"/><i><font color="Green" title="E107">A699</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">a1</font> &amp; <font color="Maroon" title="c25">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">a2</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2"><i><font color="Green" title="E102">A694</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_28_3_3_2"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_28_3_3_2_1_1"/>
( ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_2"><i><font color="Green" title="E106">A698</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_2_1_1_1"/><i><font color="Green" title="E108">A700</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_1"/><b>then </b><i><font color="Green" title="E109">A701</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2"><i><font color="Green" title="E104">A696</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2"><i><font color="Green" title="E105">A697</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_2_1_1_1"/><b>then </b><i><font color="Green" title="E110">A702</font></i>: 
<font color="Maroon" title="c24">q2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2"><i><font color="Green" title="E101">A693</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c25">a1</font>,<font color="Maroon" title="c26">a2</font>
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2"><i><font color="Green" title="E102">A694</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green" title="E103">A695</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2"><i><font color="Green" title="E104">A696</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2"><i><font color="Green" title="E105">A697</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2_1_1_1"><i><font color="Green" title="E108">A700</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2"><i><font color="Green" title="E101">A693</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_1"><i><font color="Green" title="E109">A701</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2_1_1_1"><i><font color="Green" title="E110">A702</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>
;<br/><b>hence </b><a NAME="E5:1_1_1_28_3_3_2_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green" title="E89">A655</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_2_1_1_2"/>
( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E108">A703</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2"><i><font color="Green" title="E104">A696</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2"><i><font color="Green" title="E105">A697</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E109">A704</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/><a NAME="E4:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E110">A705</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a></i>;<br/><a NAME="E5:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E111">A706</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E112">A707</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green" title="E28">A321</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a></i>;<br/><a NAME="E7:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E113">A708</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c21">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><i><font color="Green" title="E114">A709</font></i>: <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E111">A706</font></i></a></i>
;<br/><a NAME="E9:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E115">A710</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a></i>;<br/> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c21">p1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2"><i><font color="Green" title="E101">A693</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2"><i><font color="Green" title="E102">A694</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green" title="E103">A695</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E108">A703</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E109">A704</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E110">A705</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E114">A709</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E115">A710</font></i></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></i>
;<br/><a NAME="E11:1_1_1_28_3_3_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green" title="E29">A322</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green" title="E67">A606</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green" title="E89">A655</font></i></a></i>;<br/><a NAME="E12:1_1_1_28_3_3_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E112">A707</font></i></a>, <a class="ref" href="gobrd13.html#T44" title="GOBRD13:th.44">GOBRD13:44</a></i>;<br/><a NAME="E13:1_1_1_28_3_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E116">A711</font></i>: 
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">p1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2"><i><font color="Green" title="E102">A694</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green" title="E103">A695</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E108">A703</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E113">A708</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><b>set </b><font color="Maroon" title="c29">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>;<br/><a NAME="E14:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E117">A712</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green" title="E70">A636</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green" title="E72">A638</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green" title="E73">A639</font></i></a>, <a class="ref" href="jordan9.html#T29" title="JORDAN9:th.29">JORDAN9:29</a></i>;<br/><a NAME="E15:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E118">A713</font></i>: 
<font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E108">A703</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E16:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E119">A714</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E75:1_1_1"><i><font color="Green" title="E57">A556</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:1_1_1_28_3_3_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E22:1_1_1_28_3"><i><font color="Green" title="E85">A651</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E120">A715</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E119">A714</font></i></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E119">A714</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/><a NAME="E19:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E121">A716</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">a2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_2"><i><font color="Green" title="E107">A699</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E20:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E122">A717</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">a2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E108">A703</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E116">A711</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green" title="E38">A391</font></i></a>, <a class="txt" href="jordan13.html#E64:1_1_1"><i><font color="Green" title="E48">A401</font></i></a>, <a class="txt" href="jordan13.html#E66:1_1_1"><i><font color="Green" title="E49">A402</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E22:1_1_1_28_3_3_2_1_1_2"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c25">a1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">a2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green" title="E70">A636</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green" title="E72">A638</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green" title="E73">A639</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E108">A703</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E116">A711</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E118">A713</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E120">A715</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E121">A716</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/><a NAME="E23:1_1_1_28_3_3_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1_28_3"><i><font color="Green" title="E100">A666</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green" title="E103">A695</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_2"><i><font color="Green" title="E107">A699</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E108">A703</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E121">A716</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E122">A717</font></i></a>, <a class="ref" href="jordan9.html#T22" title="JORDAN9:th.22">JORDAN9:22</a></i>;<br/><a NAME="E24:1_1_1_28_3_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E123">A718</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1_28_3"><i><font color="Green" title="E90">A656</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E25:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green" title="E124">A719</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c11">f</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E86:1_1_1"><i><font color="Green" title="E63">A574</font></i></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/><a NAME="E26:1_1_1_28_3_3_2_1_1_2"/>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></i>;<br/><b>hence </b><a NAME="E27:1_1_1_28_3_3_2_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E88:1_1_1"><i><font color="Green" title="E65">A576</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_28_3"><i><font color="Green" title="E90">A656</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E117">A712</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E123">A718</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3_3_2_1_1_2"><i><font color="Green" title="E124">A719</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_2_1_1_3"/>
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_3"/><b>then </b><i><font color="Green" title="E108">A720</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2"><i><font color="Green" title="E101">A693</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2"><i><font color="Green" title="E104">A696</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2"><i><font color="Green" title="E105">A697</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green" title="E92">A658</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green" title="E103">A695</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_3"><i><font color="Green" title="E108">A720</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green" title="E82">A648</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1_28_3_3_2_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green" title="E35">A388</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green" title="E82">A648</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E32:1_1_1_28_3"><i><font color="Green" title="E93">A659</font></i></a>, <a class="txt" href="jordan13.html#E35:1_1_1_28_3"><i><font color="Green" title="E94">A660</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_2_1_1_4"/>
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_4"/><b>then </b><i><font color="Green" title="E108">A721</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2"><i><font color="Green" title="E104">A696</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2"><i><font color="Green" title="E105">A697</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c21">p1</font>,<font color="Maroon" title="c22">p2</font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2"><i><font color="Green" title="E101">A693</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2"><i><font color="Green" title="E102">A694</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green" title="E103">A695</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_4"><i><font color="Green" title="E108">A721</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1_28_3_3_2_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green" title="E81">A647</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green" title="E88">A654</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_3"/><i><font color="Green" title="E101">A722</font></i>: 
( <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c25">a1</font>, <font color="Maroon" title="c26">a2</font>, <font color="Maroon" title="c27">p'1</font>, <font color="Maroon" title="c28">p'2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_28_3_3_3"/><i><font color="Green" title="E102">A723</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">a1</font>,<font color="Maroon" title="c26">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_28_3_3_3"/><i><font color="Green" title="E103">A724</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">a1</font>,<font color="Maroon" title="c26">a2</font>
 <b>and </b><br/><a NAME="E5:1_1_1_28_3_3_3"/><i><font color="Green" title="E104">A725</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">p'1</font>,<font color="Maroon" title="c28">p'2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_28_3_3_3"/><i><font color="Green" title="E105">A726</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">p'1</font>,<font color="Maroon" title="c28">p'2</font>
 <b>and </b><br/><a NAME="E7:1_1_1_28_3_3_3"/><i><font color="Green" title="E106">A727</font></i>: 
( ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_28_3_3_3"/><i><font color="Green" title="E107">A728</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">a1</font> &amp; <font color="Maroon" title="c25">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">a2</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3"><i><font color="Green" title="E102">A723</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_28_3_3_3"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_28_3_3_3_1_1"/>
( ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_3"><i><font color="Green" title="E106">A727</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_3_1_1_1"/><i><font color="Green" title="E108">A729</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_1"/><b>then </b><i><font color="Green" title="E109">A730</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3"><i><font color="Green" title="E104">A725</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3"><i><font color="Green" title="E105">A726</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_3_1_1_1"/><b>then </b><i><font color="Green" title="E110">A731</font></i>: 
<font color="Maroon" title="c23">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">a1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green" title="E101">A722</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E4:1_1_1_28_3_3_3_1_1_1"/><i><font color="Green" title="E111">A732</font></i>: 
<font color="Maroon" title="c24">q2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green" title="E101">A722</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_3_1_1_1"><i><font color="Green" title="E109">A730</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c23">q1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">q2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green" title="E101">A722</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3"><i><font color="Green" title="E102">A723</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green" title="E103">A724</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3"><i><font color="Green" title="E104">A725</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3"><i><font color="Green" title="E105">A726</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3_1_1_1"><i><font color="Green" title="E108">A729</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3_1_1_1"><i><font color="Green" title="E110">A731</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3_1_1_1"><i><font color="Green" title="E111">A732</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>
;<br/><b>hence </b><a NAME="E6:1_1_1_28_3_3_3_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green" title="E81">A647</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green" title="E88">A654</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_3_1_1_2"/>
( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_2"/><b>then </b><i><font color="Green" title="E108">A733</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3"><i><font color="Green" title="E104">A725</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3"><i><font color="Green" title="E105">A726</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green" title="E92">A658</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green" title="E101">A722</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green" title="E103">A724</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_3_1_1_2"><i><font color="Green" title="E108">A733</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green" title="E82">A648</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1_28_3_3_3_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green" title="E35">A388</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green" title="E82">A648</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E32:1_1_1_28_3"><i><font color="Green" title="E93">A659</font></i></a>, <a class="txt" href="jordan13.html#E35:1_1_1_28_3"><i><font color="Green" title="E94">A660</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_3_1_1_3"/>
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_3"/><b>then </b><i><font color="Green" title="E108">A734</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3"><i><font color="Green" title="E104">A725</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3"><i><font color="Green" title="E105">A726</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green" title="E109">A735</font></i>: 
<font color="Maroon" title="c21">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green" title="E101">A722</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E4:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green" title="E110">A736</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/><a NAME="E5:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green" title="E111">A737</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a></i>;<br/><a NAME="E6:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green" title="E112">A738</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E7:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green" title="E113">A739</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green" title="E28">A321</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a></i>;<br/><a NAME="E8:1_1_1_28_3_3_3_1_1_3"/>
<font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c21">p1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E9:1_1_1_28_3_3_3_1_1_3"/><b>then </b><i><font color="Green" title="E114">A740</font></i>: 
( <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c22">p2</font> &amp; <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E108">A734</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E115">A741</font></i>: <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E112">A738</font></i></a></i>
;<br/><a NAME="E11:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green" title="E116">A742</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a></i>;<br/> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c23">q1</font>,<font color="Maroon" title="c24">q2</font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green" title="E101">A722</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green" title="E101">A722</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3"><i><font color="Green" title="E102">A723</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green" title="E103">A724</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E108">A734</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E109">A735</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E110">A736</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E111">A737</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E115">A741</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E116">A742</font></i></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></i>
;<br/><a NAME="E13:1_1_1_28_3_3_3_1_1_3"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green" title="E29">A322</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green" title="E67">A606</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green" title="E89">A655</font></i></a></i>;<br/><a NAME="E14:1_1_1_28_3_3_3_1_1_3"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E113">A739</font></i></a>, <a class="ref" href="gobrd13.html#T44" title="GOBRD13:th.44">GOBRD13:44</a></i>;<br/><a NAME="E15:1_1_1_28_3_3_3_1_1_3"/><b>then </b><i><font color="Green" title="E117">A743</font></i>: 
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c21">p1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3"><i><font color="Green" title="E102">A723</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green" title="E103">A724</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E114">A740</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><a NAME="E16:1_1_1_28_3_3_3_1_1_3"/><b>then </b><i><font color="Green" title="E118">A744</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>set </b><font color="Maroon" title="c29">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>;<br/><a NAME="E17:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green" title="E119">A745</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green" title="E70">A636</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green" title="E72">A638</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green" title="E73">A639</font></i></a>, <a class="ref" href="jordan9.html#T29" title="JORDAN9:th.29">JORDAN9:29</a></i>;<br/><a NAME="E18:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green" title="E120">A746</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E75:1_1_1"><i><font color="Green" title="E57">A556</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E19:1_1_1_28_3_3_3_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E22:1_1_1_28_3"><i><font color="Green" title="E85">A651</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E121">A747</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E18:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E120">A746</font></i></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E120">A746</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green" title="E38">A391</font></i></a>, <a class="txt" href="jordan13.html#E64:1_1_1"><i><font color="Green" title="E48">A401</font></i></a>, <a class="txt" href="jordan13.html#E66:1_1_1"><i><font color="Green" title="E49">A402</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E22:1_1_1_28_3_3_3_1_1_3"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c21">p1</font>,<font color="Maroon" title="c22">p2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green" title="E70">A636</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green" title="E72">A638</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green" title="E73">A639</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E117">A743</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E121">A747</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/><a NAME="E23:1_1_1_28_3_3_3_1_1_3"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1_28_3"><i><font color="Green" title="E100">A666</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green" title="E103">A724</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_3"><i><font color="Green" title="E107">A728</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E108">A734</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E118">A744</font></i></a>, <a class="ref" href="jordan9.html#T22" title="JORDAN9:th.22">JORDAN9:22</a></i>;<br/><a NAME="E24:1_1_1_28_3_3_3_1_1_3"/><b>then </b><i><font color="Green" title="E122">A748</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1_28_3"><i><font color="Green" title="E90">A656</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E25:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green" title="E123">A749</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c11">f</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E86:1_1_1"><i><font color="Green" title="E63">A574</font></i></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/><a NAME="E26:1_1_1_28_3_3_3_1_1_3"/>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></i>;<br/><b>hence </b><a NAME="E27:1_1_1_28_3_3_3_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E88:1_1_1"><i><font color="Green" title="E65">A576</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_28_3"><i><font color="Green" title="E90">A656</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E119">A745</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E122">A748</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3_3_3_1_1_3"><i><font color="Green" title="E123">A749</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_3_1_1_4"/>
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_4"/><b>then </b><i><font color="Green" title="E108">A750</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3"><i><font color="Green" title="E104">A725</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3"><i><font color="Green" title="E105">A726</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_3_1_1_4"/><b>then </b>
<font color="Maroon" title="c23">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">a1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green" title="E101">A722</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><b>then </b> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c23">q1</font>,<font color="Maroon" title="c24">q2</font>
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green" title="E101">A722</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3"><i><font color="Green" title="E102">A723</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green" title="E103">A724</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_3_1_1_4"><i><font color="Green" title="E108">A750</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green" title="E101">A722</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>
;<br/><b>hence </b><a NAME="E5:1_1_1_28_3_3_3_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green" title="E89">A655</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_4"/><i><font color="Green" title="E101">A751</font></i>: 
( <font color="Maroon" title="c21">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> &amp; <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c25">a1</font>, <font color="Maroon" title="c26">a2</font>, <font color="Maroon" title="c27">p'1</font>, <font color="Maroon" title="c28">p'2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:1_1_1_28_3_3_4"/><i><font color="Green" title="E102">A752</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">a1</font>,<font color="Maroon" title="c26">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_28_3_3_4"/><i><font color="Green" title="E103">A753</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c25">a1</font>,<font color="Maroon" title="c26">a2</font>
 <b>and </b><br/><a NAME="E5:1_1_1_28_3_3_4"/><i><font color="Green" title="E104">A754</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">p'1</font>,<font color="Maroon" title="c28">p'2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_28_3_3_4"/><i><font color="Green" title="E105">A755</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">p'1</font>,<font color="Maroon" title="c28">p'2</font>
 <b>and </b><br/><a NAME="E7:1_1_1_28_3_3_4"/><i><font color="Green" title="E106">A756</font></i>: 
( ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_28_3_3_4"/><i><font color="Green" title="E107">A757</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">a1</font> &amp; <font color="Maroon" title="c25">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">a2</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4"><i><font color="Green" title="E102">A752</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_28_3_3_4"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_28_3_3_4_1_1"/>
( ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> ) or ( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_4"><i><font color="Green" title="E106">A756</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_4_1_1_1"/>
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_1"/><b>then </b><i><font color="Green" title="E108">A758</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4"><i><font color="Green" title="E104">A754</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4"><i><font color="Green" title="E105">A755</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green" title="E92">A658</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green" title="E101">A751</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green" title="E103">A753</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_4_1_1_1"><i><font color="Green" title="E108">A758</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green" title="E82">A648</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1_28_3_3_4_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green" title="E35">A388</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green" title="E82">A648</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E32:1_1_1_28_3"><i><font color="Green" title="E93">A659</font></i></a>, <a class="txt" href="jordan13.html#E35:1_1_1_28_3"><i><font color="Green" title="E94">A660</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_4_1_1_2"/><i><font color="Green" title="E108">A759</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_2"/><b>then </b><i><font color="Green" title="E109">A760</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4"><i><font color="Green" title="E104">A754</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4"><i><font color="Green" title="E105">A755</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_4_1_1_2"/><b>then </b><i><font color="Green" title="E110">A761</font></i>: 
<font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green" title="E101">A751</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E4:1_1_1_28_3_3_4_1_1_2"/><i><font color="Green" title="E111">A762</font></i>: 
<font color="Maroon" title="c26">a2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green" title="E101">A751</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_4_1_1_2"><i><font color="Green" title="E109">A760</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c25">a1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">a2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4"><i><font color="Green" title="E102">A752</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green" title="E103">A753</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4"><i><font color="Green" title="E104">A754</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4"><i><font color="Green" title="E105">A755</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4_1_1_2"><i><font color="Green" title="E108">A759</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green" title="E101">A751</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4_1_1_2"><i><font color="Green" title="E110">A761</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4_1_1_2"><i><font color="Green" title="E111">A762</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>
;<br/><b>hence </b><a NAME="E6:1_1_1_28_3_3_4_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green" title="E89">A655</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_4_1_1_3"/>
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_3"/><b>then </b><i><font color="Green" title="E108">A763</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4"><i><font color="Green" title="E104">A754</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4"><i><font color="Green" title="E105">A755</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_4_1_1_3"/><b>then </b><i><font color="Green" title="E109">A764</font></i>: 
<font color="Maroon" title="c26">a2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green" title="E101">A751</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c23">q1</font>,<font color="Maroon" title="c24">q2</font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green" title="E101">A751</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green" title="E101">A751</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4"><i><font color="Green" title="E102">A752</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green" title="E103">A753</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_4_1_1_3"><i><font color="Green" title="E108">A763</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4_1_1_3"><i><font color="Green" title="E109">A764</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>
;<br/><b>hence </b><a NAME="E5:1_1_1_28_3_3_4_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green" title="E81">A647</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green" title="E88">A654</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_4_1_1_4"/>
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p'1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p'2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_4"/><b>then </b><i><font color="Green" title="E108">A765</font></i>: 
( <font color="Maroon" title="c25">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font> &amp; <font color="Maroon" title="c26">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4"><i><font color="Green" title="E104">A754</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4"><i><font color="Green" title="E105">A755</font></i></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E109">A766</font></i>: 
<font color="Maroon" title="c22">p2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green" title="E101">A751</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/><a NAME="E4:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E110">A767</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/><a NAME="E5:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E111">A768</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green" title="E13">A92</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a></i>;<br/><a NAME="E6:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E112">A769</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E7:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E113">A770</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green" title="E28">A321</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a></i>;<br/><a NAME="E8:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E114">A771</font></i>: 
( <font color="Maroon" title="c26">a2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c22">p2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c22">p2</font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E108">A765</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E115">A772</font></i>: <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E59:1_1_1"><i><font color="Green" title="E46">A399</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E112">A769</font></i></a></i>
;<br/><a NAME="E10:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E116">A773</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">m</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a></i>;<br/> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c11">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">q1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c24">q2</font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green" title="E80">A646</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3"><i><font color="Green" title="E83">A649</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green" title="E86">A652</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green" title="E87">A653</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green" title="E97">A663</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green" title="E98">A664</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green" title="E101">A751</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green" title="E101">A751</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4"><i><font color="Green" title="E102">A752</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green" title="E103">A753</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E108">A765</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E109">A766</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E110">A767</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E111">A768</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E115">A772</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E116">A773</font></i></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></i>
;<br/><a NAME="E12:1_1_1_28_3_3_4_1_1_4"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green" title="E29">A322</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green" title="E67">A606</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green" title="E89">A655</font></i></a></i>;<br/><a NAME="E13:1_1_1_28_3_3_4_1_1_4"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E56:1_1_1"><i><font color="Green" title="E45">A398</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E113">A770</font></i></a>, <a class="ref" href="gobrd13.html#T44" title="GOBRD13:th.44">GOBRD13:44</a></i>;<br/><a NAME="E14:1_1_1_28_3_3_4_1_1_4"/><b>then </b><i><font color="Green" title="E117">A774</font></i>: 
( <font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">p2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green" title="E74">A640</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4"><i><font color="Green" title="E102">A752</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green" title="E103">A753</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E114">A771</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><b>set </b><font color="Maroon" title="c29">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>;<br/><a NAME="E15:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E118">A775</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green" title="E70">A636</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green" title="E72">A638</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green" title="E73">A639</font></i></a>, <a class="ref" href="jordan9.html#T29" title="JORDAN9:th.29">JORDAN9:29</a></i>;<br/><a NAME="E16:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E119">A776</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E75:1_1_1"><i><font color="Green" title="E57">A556</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:1_1_1_28_3_3_4_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E22:1_1_1_28_3"><i><font color="Green" title="E85">A651</font></i></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E120">A777</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E119">A776</font></i></a>, <a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green" title="E91">A657</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E119">A776</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/><i><font color="Green" title="E121">A778</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c14">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green" title="E38">A391</font></i></a>, <a class="txt" href="jordan13.html#E64:1_1_1"><i><font color="Green" title="E48">A401</font></i></a>, <a class="txt" href="jordan13.html#E66:1_1_1"><i><font color="Green" title="E49">A402</font></i></a>, <a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T68" title="FINSEQ_5:th.68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E20:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E122">A779</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E117">A774</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E21:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E123">A780</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p1</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1_28_3"><i><font color="Green" title="E100">A666</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E22:1_1_1_28_3_3_4_1_1_4"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">p1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c22">p2</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green" title="E70">A636</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green" title="E72">A638</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green" title="E73">A639</font></i></a>, <a class="txt" href="jordan13.html#E37:1_1_1_28_3"><i><font color="Green" title="E95">A661</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green" title="E96">A662</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E117">A774</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E120">A777</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E121">A778</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/><a NAME="E23:1_1_1_28_3_3_4_1_1_4"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1_28_3"><i><font color="Green" title="E100">A666</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green" title="E103">A753</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_4"><i><font color="Green" title="E107">A757</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E108">A765</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E122">A779</font></i></a>, <a class="txt" href="jordan13.html#E21:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E123">A780</font></i></a>, <a class="ref" href="jordan9.html#T22" title="JORDAN9:th.22">JORDAN9:22</a></i>;<br/><a NAME="E24:1_1_1_28_3_3_4_1_1_4"/><b>then </b><i><font color="Green" title="E124">A781</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1_28_3"><i><font color="Green" title="E90">A656</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E25:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green" title="E125">A782</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c11">f</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E86:1_1_1"><i><font color="Green" title="E63">A574</font></i></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/><a NAME="E26:1_1_1_28_3_3_4_1_1_4"/>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></i>;<br/><b>hence </b><a NAME="E27:1_1_1_28_3_3_4_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green" title="E41">A394</font></i></a>, <a class="txt" href="jordan13.html#E88:1_1_1"><i><font color="Green" title="E65">A576</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green" title="E75">A641</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green" title="E76">A642</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_28_3"><i><font color="Green" title="E90">A656</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E118">A775</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E124">A781</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3_3_4_1_1_4"><i><font color="Green" title="E125">A782</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:1_1_1_28"/><b>then </b>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="ref" href="goboard9.html#T26" title="GOBOARD9:th.26">GOBOARD9:26</a></i>;<br/>
<a NAME="E7:1_1_1_28"/><b>then </b>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E81:1_1_1"><i><font color="Green" title="E60">A564</font></i></a>, <a class="txt" href="jordan13.html#E87:1_1_1"><i><font color="Green" title="E64">A575</font></i></a>, <a class="txt" href="jordan13.html#E89:1_1_1"><i><font color="Green" title="E66">A595</font></i></a>, <a class="ref" href="jordan9.html#T3" title="JORDAN9:th.3">JORDAN9:3</a></i>;<br/>
<b>hence </b><a NAME="E8:1_1_1_28"/>
contradiction
 <b>by </b><i><a class="ref" href="sprect_4.html#T7" title="SPRECT_4:th.7">SPRECT_4:7</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E67">A783</font></i>: <font color="Maroon" title="c14">g</font> = 
<font color="Maroon" title="c11">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 0
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font>
<b>by </b><i><a class="ref" href="finseq_5.html#T31" title="FINSEQ_5:th.31">FINSEQ_5:31</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c11">f</font> as  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a> ;<br/>
<a NAME="E93:1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E68">A784</font></i>: <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T25" title="FINSEQ_4:th.25">FINSEQ_4:25</a></i>
<br/>.= 
<font color="Maroon" title="c16">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E41:1_1_1"><i><font color="Green" title="E34">A387</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a></i>
;<br/>
<a NAME="E95:1_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green" title="E6">A78</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E69">A785</font></i>: <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 2</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>
<br/>.= 
<font color="Maroon" title="c16">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E54:1_1_1"><i><font color="Green" title="E44">A397</font></i></a>, <a class="txt" href="jordan13.html#E79:1_1_1"><i><font color="Green" title="E59">A560</font></i></a></i>
;<br/>
<a NAME="E97:1_1_1"/>
<font color="Maroon" title="c16">f</font> is <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_32"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_32"/><i><font color="Green" title="E70">A786</font></i>: 
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c16">f</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></i>;<br/>
<a NAME="E2:1_1_1_32"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">f</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T73" title="JORDAN2C:th.73">JORDAN2C:73</a></i>;<br/>
<a NAME="E3:1_1_1_32"/><b>then </b>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">f</font></span>)</span> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">f</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T76" title="JORDAN2C:th.76">JORDAN2C:76</a></i>;<br/>
<a NAME="E4:1_1_1_32"/><b>then </b><i><font color="Green" title="E71">A787</font></i>: 
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">f</font></span>)</span> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#D4" title="JORDAN2C:def.4">JORDAN2C:def 4</a></i>;<br/>
<a NAME="E5:1_1_1_32"/><i><font color="Green" title="E72">A788</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c16">f</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c16">f</font>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<a NAME="E6:1_1_1_32"/><i><font color="Green" title="E73">A789</font></i>: 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="jordan11.html#T6" title="JORDAN11:th.6">JORDAN11:6</a></i>;<br/>
<a NAME="E7:1_1_1_32"/><i><font color="Green" title="E74">A790</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1"><i><font color="Green" title="E14">A93</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E8:1_1_1_32"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="jordan11.html#T9" title="JORDAN11:th.9">JORDAN11:9</a></i>;<br/>
<a NAME="E9:1_1_1_32"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c16">f</font>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E19:1_1_1"><i><font color="Green" title="E16">A95</font></i></a>, <a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E53:1_1_1"><i><font color="Green" title="E43">A396</font></i></a>, <a class="txt" href="jordan13.html#E94:1_1_1"><i><font color="Green" title="E68">A784</font></i></a>, <a class="txt" href="jordan13.html#E96:1_1_1"><i><font color="Green" title="E69">A785</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_32"><i><font color="Green" title="E73">A789</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_32"><i><font color="Green" title="E74">A790</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/>
<a NAME="E10:1_1_1_32"/><b>then </b><i><font color="Green" title="E75">A791</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c16">f</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_32"><i><font color="Green" title="E72">A788</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:1_1_1_32"/><i><font color="Green" title="E76">A792</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c16">f</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c16">f</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E53:1_1_1"><i><font color="Green" title="E43">A396</font></i></a>, <a class="ref" href="jordan1h.html#T31" title="JORDAN1H:th.31">JORDAN1H:31</a></i>;<br/>
<a NAME="E12:1_1_1_32"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c16">f</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E53:1_1_1"><i><font color="Green" title="E43">A396</font></i></a>, <a class="ref" href="jordan9.html#T11" title="JORDAN9:th.11">JORDAN9:11</a></i>;<br/>
<a NAME="E13:1_1_1_32"/><b>then </b><i><font color="Green" title="E77">A793</font></i>: 
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c16">f</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_32"><i><font color="Green" title="E75">A791</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_32"><i><font color="Green" title="E76">A792</font></i></a>, <a class="ref" href="xboole_1.html#T68" title="XBOOLE_1:th.68">XBOOLE_1:68</a></i>;<br/>
<a NAME="E14:1_1_1_32"/><i><font color="Green" title="E78">A794</font></i>: 
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c16">f</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E81:1_1_1"><i><font color="Green" title="E60">A564</font></i></a>, <a class="txt" href="jordan13.html#E89:1_1_1"><i><font color="Green" title="E66">A595</font></i></a>, <a class="txt" href="jordan13.html#E91:1_1_1"><i><font color="Green" title="E67">A783</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_32"><i><font color="Green" title="E70">A786</font></i></a>, <a class="ref" href="goboard9.html#T6" title="GOBOARD9:th.6">GOBOARD9:6</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">W</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">C</font> </span>}</span> ;<br/>
<a NAME="E15:1_1_1_32"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">C</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#D5" title="JORDAN2C:def.5">JORDAN2C:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:1_1_1_32"/><i><font color="Green" title="E79">A795</font></i>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">C</font> </span>}</span> 
 <b>and </b><br/><a NAME="E18:1_1_1_32"/><i><font color="Green" title="E80">A796</font></i>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c16">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c18">e</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E13:1_1_1_32"><i><font color="Green" title="E77">A793</font></i></a>, <a class="ref" href="zfmisc_1.html#T98" title="ZFMISC_1:th.98">ZFMISC_1:98</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E20:1_1_1_32"/><i><font color="Green" title="E81">A797</font></i>: 
<font color="Maroon" title="c18">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">B</font>
 <b>and </b><br/><a NAME="E21:1_1_1_32"/><i><font color="Green" title="E82">A798</font></i>: 
<font color="Maroon" title="c19">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1_32"><i><font color="Green" title="E79">A795</font></i></a></i>;<br/>
<a NAME="E22:1_1_1_32"/><i><font color="Green" title="E83">A799</font></i>: 
<font color="Maroon" title="c19">B</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan13.html#E21:1_1_1_32"><i><font color="Green" title="E82">A798</font></i></a>, <a class="ref" href="jordan2c.html#D3" title="JORDAN2C:def.3">JORDAN2C:def 3</a></i>;<br/>
<a NAME="E23:1_1_1_32"/>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c16">f</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c16">f</font>
 
<b>by </b><i><a class="ref" href="gobrd14.html#T24" title="GOBRD14:th.24">GOBRD14:24</a></i>;<br/>
<a NAME="E24:1_1_1_32"/><b>then </b>
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c16">f</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_32"><i><font color="Green" title="E78">A794</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E25:1_1_1_32"/><b>then </b>
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c16">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>
<a NAME="E26:1_1_1_32"/><b>then </b><i><font color="Green" title="E84">A800</font></i>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c16">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">B</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E18:1_1_1_32"><i><font color="Green" title="E80">A796</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_32"><i><font color="Green" title="E81">A797</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_32"><i><font color="Green" title="E83">A799</font></i></a>, <a class="ref" href="goboard9.html#T6" title="GOBOARD9:th.6">GOBOARD9:6</a></i>;<br/>
<a NAME="E27:1_1_1_32"/>
<font color="Maroon" title="c19">B</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 
<b>by </b><i><a class="txt" href="jordan13.html#E21:1_1_1_32"><i><font color="Green" title="E82">A798</font></i></a>, <a class="ref" href="jordan2c.html#D3" title="JORDAN2C:def.3">JORDAN2C:def 3</a></i>;<br/>
<a NAME="E28:1_1_1_32"/><b>then </b><i><font color="Green" title="E85">A801</font></i>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c16">f</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 
<b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1_32"><i><font color="Green" title="E84">A800</font></i></a>, <a class="ref" href="jordan2c.html#T16" title="JORDAN2C:th.16">JORDAN2C:16</a></i>;<br/>
<a NAME="E29:1_1_1_32"/><i><font color="Green" title="E86">A802</font></i>: 
(  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c16">f</font> or  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c16">f</font> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_32"><i><font color="Green" title="E71">A787</font></i></a>, <a class="ref" href="jordan1h.html#T30" title="JORDAN1H:th.30">JORDAN1H:30</a></i>;<br/>
<a NAME="E30:1_1_1_32"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c16">f</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T73" title="JORDAN2C:th.73">JORDAN2C:73</a></i>;<br/>
<b>hence </b><a NAME="E31:1_1_1_32"/>
<font color="Maroon" title="c16">f</font> is <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1_32"><i><font color="Green" title="E85">A801</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_32"><i><font color="Green" title="E86">A802</font></i></a>, <a class="ref" href="jordan1h.html#T47" title="JORDAN1H:th.47">JORDAN1H:47</a>, <a class="ref" href="jordan1h.html#T49" title="JORDAN1H:th.49">JORDAN1H:49</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c17">f</font> = <font color="Maroon" title="c16">f</font> as  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c17">f</font>
;<br/>

<b>thus </b><a NAME="E99:1_1_1"/>
<font color="Maroon" title="c17">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green" title="E19">A245</font></i></a></i>;<br/>

<b>thus </b><a NAME="E100:1_1_1"/>
<font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E94:1_1_1"><i><font color="Green" title="E68">A784</font></i></a></i>;<br/>

<b>thus </b><a NAME="E101:1_1_1"/>
<font color="Maroon" title="c17">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E96:1_1_1"><i><font color="Green" title="E69">A785</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c18">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E102:1_1_1"/><i><font color="Green" title="E70">A803</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">m</font> &amp; <font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">f</font> )
 ;<br/>

<a NAME="E103:1_1_1"/>
<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E104:1_1_1"/><b>then </b><i><font color="Green" title="E71">A804</font></i>: 
<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">f</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E102:1_1_1"><i><font color="Green" title="E70">A803</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E105:1_1_1"/><b>then </b><i><font color="Green" title="E72">A805</font></i>: 
<font color="Maroon" title="c17">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green" title="E28">A321</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a></i>;<br/>
<a NAME="E106:1_1_1"/><i><font color="Green" title="E73">A806</font></i>: 
<font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>;<br/>
<a NAME="E107:1_1_1"/><i><font color="Green" title="E74">A807</font></i>: 
<font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E102:1_1_1"><i><font color="Green" title="E70">A803</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E108:1_1_1"/><i><font color="Green" title="E75">A808</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c18">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c17">f</font>,<font color="Maroon" title="c18">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c18">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c17">f</font>,<font color="Maroon" title="c18">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E49:1_1_1"><i><font color="Green" title="E40">A393</font></i></a>, <a class="txt" href="jordan13.html#E102:1_1_1"><i><font color="Green" title="E70">A803</font></i></a>, <a class="txt" href="jordan13.html#E104:1_1_1"><i><font color="Green" title="E71">A804</font></i></a>, <a class="txt" href="jordan13.html#E105:1_1_1"><i><font color="Green" title="E72">A805</font></i></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></i>;<br/>
<a NAME="E109:1_1_1"/><i><font color="Green" title="E76">A809</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green" title="E28">A321</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green" title="E39">A392</font></i></a>, <a class="txt" href="jordan13.html#E102:1_1_1"><i><font color="Green" title="E70">A803</font></i></a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:1_1_1_33"/>
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c17">f</font>,<font color="Maroon" title="c18">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c17">f</font>,<font color="Maroon" title="c18">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><a NAME="E2:1_1_1_33"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c18">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green" title="E29">A322</font></i></a>, <a class="txt" href="jordan13.html#E106:1_1_1"><i><font color="Green" title="E73">A806</font></i></a>, <a class="txt" href="jordan13.html#E107:1_1_1"><i><font color="Green" title="E74">A807</font></i></a>, <a class="txt" href="jordan13.html#E108:1_1_1"><i><font color="Green" title="E75">A808</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_33"/>
<font color="Maroon" title="c17">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c18">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E102:1_1_1"><i><font color="Green" title="E70">A803</font></i></a>, <a class="txt" href="jordan13.html#E109:1_1_1"><i><font color="Green" title="E76">A809</font></i></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:1_1_1_34"/>
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c17">f</font>,<font color="Maroon" title="c18">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c17">f</font>,<font color="Maroon" title="c18">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><a NAME="E2:1_1_1_34"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c18">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green" title="E29">A322</font></i></a>, <a class="txt" href="jordan13.html#E106:1_1_1"><i><font color="Green" title="E73">A806</font></i></a>, <a class="txt" href="jordan13.html#E107:1_1_1"><i><font color="Green" title="E74">A807</font></i></a>, <a class="txt" href="jordan13.html#E108:1_1_1"><i><font color="Green" title="E75">A808</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_34"/>
<font color="Maroon" title="c17">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c18">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E102:1_1_1"><i><font color="Green" title="E70">A803</font></i></a>, <a class="txt" href="jordan13.html#E109:1_1_1"><i><font color="Green" title="E76">A809</font></i></a>, <a class="ref" href="gobrd13.html#T46" title="GOBRD13:th.46">GOBRD13:46</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E112:1_1_1"/>
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c17">f</font>,<font color="Maroon" title="c18">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<a NAME="E113:1_1_1"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c18">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green" title="E29">A322</font></i></a>, <a class="txt" href="jordan13.html#E106:1_1_1"><i><font color="Green" title="E73">A806</font></i></a>, <a class="txt" href="jordan13.html#E107:1_1_1"><i><font color="Green" title="E74">A807</font></i></a>, <a class="txt" href="jordan13.html#E108:1_1_1"><i><font color="Green" title="E75">A808</font></i></a></i>;<br/>
<b>hence </b><a NAME="E114:1_1_1"/>
<font color="Maroon" title="c17">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c18">m</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan13.html#E102:1_1_1"><i><font color="Green" title="E70">A803</font></i></a>, <a class="txt" href="jordan13.html#E109:1_1_1"><i><font color="Green" title="E76">A809</font></i></a>, <a class="ref" href="gobrd13.html#T44" title="GOBRD13:th.44">GOBRD13:44</a></i>;<br/>


</div>
