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<b>let </b><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</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="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>assume </b><a NAME="E1:33_1_2"/><i><font color="Green" title="E31">A832</font></i>: 
<font color="Maroon" title="c3">f1</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>
 ;<br/>

<b>given </b><font color="Maroon" title="c5">i1</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><a NAME="E3:33_1_2"/><i><font color="Green" title="E32">A833</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i1</font> &amp; <font color="Maroon" title="c5">i1</font> <a href="nat_1.html#K2" title="NAT_1: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> <span class="p1">(<span class="default"><a href="jordan8.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="E4:33_1_2"/><i><font color="Green" title="E33">A834</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c5">i1</font>,<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> &amp; <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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"><font color="Maroon" title="c5">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<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>and </b><br/><a NAME="E5:33_1_2"/><i><font color="Green" title="E34">A835</font></i>: 
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">i1</font>,<span class="p1">(<span class="default"><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> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E6:33_1_2"/><i><font color="Green" title="E35">A836</font></i>: 
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i1</font>,<span class="p1">(<span class="default"><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> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E7:33_1_2"/><i><font color="Green" title="E36">A837</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c3">f1</font> holds <br/>( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">k</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#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">k</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> implies <font color="Maroon" title="c3">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</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> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">k</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#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">k</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> implies <font color="Maroon" title="c3">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</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> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f1</font>,<font color="Olive" title="b1">k</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> implies <font color="Maroon" title="c3">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</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>and </b><br/><a NAME="E8:33_1_2"/><i><font color="Green" title="E37">A838</font></i>: 
<font color="Maroon" title="c4">f2</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>
 ;<br/>

<b>given </b><font color="Maroon" title="c6">i2</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><a NAME="E10:33_1_2"/><i><font color="Green" title="E38">A839</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K2" title="NAT_1: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> <span class="p1">(<span class="default"><a href="jordan8.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="E11:33_1_2"/><i><font color="Green" title="E39">A840</font></i>: 
( <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c6">i2</font>,<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> &amp; <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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"><font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<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>and </b><br/><a NAME="E12:33_1_2"/><i><font color="Green" title="E40">A841</font></i>: 
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c6">i2</font>,<span class="p1">(<span class="default"><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> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E13:33_1_2"/><i><font color="Green" title="E41">A842</font></i>: 
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">N-min</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i2</font>,<span class="p1">(<span class="default"><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> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/>

<b>assume </b><a NAME="E14:33_1_2"/><i><font color="Green" title="E42">A843</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c4">f2</font> holds <br/>( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c4">f2</font>,<font color="Olive" title="b1">k</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#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c4">f2</font>,<font color="Olive" title="b1">k</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> implies <font color="Maroon" title="c4">f2</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</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> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c4">f2</font>,<font color="Olive" title="b1">k</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#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c4">f2</font>,<font color="Olive" title="b1">k</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> implies <font color="Maroon" title="c4">f2</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</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> ) &amp; (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c4">f2</font>,<font color="Olive" title="b1">k</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> implies <font color="Maroon" title="c4">f2</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</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/>

<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> ] <b>means </b><font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1;<br/>
<a NAME="E15:33_1_2"/><i><font color="Green" title="E43">A844</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E16:33_1_2"/><i><font color="Green" title="E44">A845</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>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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="E1:33_1_2_1"/><i><font color="Green" title="E45">A846</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font>
 ;<br/>

<a NAME="E2:33_1_2_1"/><i><font color="Green" title="E46">A847</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1 <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"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1 <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"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_5.html#T23" title="FINSEQ_5:th.23">FINSEQ_5:23</a></i>;<br/>
<a NAME="E3:33_1_2_1"/><i><font color="Green" title="E47">A848</font></i>: 
<font color="Maroon" title="c5">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_2"><i><font color="Green" title="E32">A833</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_2"><i><font color="Green" title="E34">A835</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_2"><i><font color="Green" title="E35">A836</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_2"><i><font color="Green" title="E38">A839</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_2"><i><font color="Green" title="E40">A841</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_2"><i><font color="Green" title="E41">A842</font></i></a>, <a class="ref" href="jordan9.html#T31" target="_self" title="JORDAN9:th.31">Th31</a></i>;<br/>
<a NAME="E4:33_1_2_1"/><i><font color="Green" title="E48">A849</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 4 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 4 )
 
<b>by </b><i><a class="ref" href="goboard7.html#T36" title="GOBOARD7:th.36">GOBOARD7:36</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_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="E1:33_1_2_1_1"/>
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">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="33_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_2_1_1_1"/>
<font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_2"><i><font color="Green" title="E32">A833</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_2"><i><font color="Green" title="E33">A834</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_2"><i><font color="Green" title="E34">A835</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_2"><i><font color="Green" title="E35">A836</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_2"><i><font color="Green" title="E38">A839</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_2"><i><font color="Green" title="E39">A840</font></i></a>, <a class="txt" href="jordan9.html#E12:33_1_2"><i><font color="Green" title="E40">A841</font></i></a>, <a class="txt" href="jordan9.html#E13:33_1_2"><i><font color="Green" title="E41">A842</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1"><i><font color="Green" title="E46">A847</font></i></a>, <a class="ref" href="jordan9.html#T31" target="_self" title="JORDAN9:th.31">Th31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_2"/><i><font color="Green" title="E49">A850</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_2"/>
( <font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2 <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"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2 <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"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</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="jordan9.html#E4:33_1_2_1"><i><font color="Green" title="E48">A849</font></i></a>, <a class="ref" href="jordan8.html#T1" title="JORDAN8:th.1">JORDAN8:1</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="E3:33_1_2_1_1_2"/>
<font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:33_1_2"><i><font color="Green" title="E33">A834</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_2"><i><font color="Green" title="E39">A840</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_2_1"><i><font color="Green" title="E47">A848</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1_1_2"><i><font color="Green" title="E49">A850</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3"/><i><font color="Green" title="E49">A851</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_3"/><i><font color="Green" title="E50">A852</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> &amp; <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_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="E1:33_1_2_1_1_3_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">k</font> or <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E51">A853</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">k</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E2:33_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E52">A854</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2_1_1_3"><i><font color="Green" title="E49">A851</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E3:33_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E53">A855</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="binarith.html#T61" title="BINARITH:th.61">BINARITH:61</a></i>;<br/><div><i><font color="Green" title="E54">A856</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_2_1_1_3_1_1_1_1"/><i><font color="Green" title="E55">A857</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 ;<br/><a NAME="E2:33_1_2_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E56">A858</font></i>: 
<font color="Maroon" title="c4">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/><a NAME="E3:33_1_2_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E57">A859</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E4:33_1_2_1_1_3_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</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="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_1_1"><i><font color="Green" title="E56">A858</font></i></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E5:33_1_2_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E58">A860</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><a NAME="E6:33_1_2_1_1_3_1_1_1_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</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="c3">f1</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_2_1"><i><font color="Green" title="E48">A849</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_1_1"><i><font color="Green" title="E56">A858</font></i></a>, <a class="ref" href="finseq_5.html#T18" title="FINSEQ_5:th.18">FINSEQ_5:18</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="E7:33_1_2_1_1_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3"><i><font color="Green" title="E50">A852</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E51">A853</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1_1_3_1_1_1_1"><i><font color="Green" title="E55">A857</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_1_1"><i><font color="Green" title="E56">A858</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_2_1_1_3_1_1_1_1"><i><font color="Green" title="E57">A859</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_2_1_1_3_1_1_1_1"><i><font color="Green" title="E58">A860</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a>, <a class="ref" href="goboard7.html#T40" title="GOBOARD7:th.40">GOBOARD7:40</a></i>;<br/></div><b>end;</b></div><a NAME="E5:33_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E55">A861</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1: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="c4">f2</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="E6:33_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E56">A862</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1: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="c3">f1</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E51">A853</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:33_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E57">A863</font></i>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2"><i><font color="Green" title="E31">A832</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E51">A853</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E52">A854</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E53">A855</font></i></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></i>;<br/><a NAME="E8:33_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E58">A864</font></i>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2"><i><font color="Green" title="E31">A832</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E51">A853</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E52">A854</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E53">A855</font></i></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></i>;<br/><a NAME="E9:33_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E59">A865</font></i>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c4">f2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_2"><i><font color="Green" title="E37">A838</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E52">A854</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E53">A855</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E54">A856</font></i></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></i>;<br/><a NAME="E10:33_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E60">A866</font></i>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c4">f2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_2"><i><font color="Green" title="E37">A838</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E52">A854</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E53">A855</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E54">A856</font></i></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></i>;<br/><a NAME="E11:33_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E61">A867</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E53">A855</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_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="E1:33_1_2_1_1_3_1_1_1_2_1"/>
( (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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> ) or (  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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> ) or  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3_1_1_1_2_1_1"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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> )
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_3_1_1_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c3">f1</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c7">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; <font color="Maroon" title="c4">f2</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c7">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="jordan9.html#E7:33_1_2"><i><font color="Green" title="E36">A837</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_2"><i><font color="Green" title="E42">A843</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E52">A854</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E55">A861</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E56">A862</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E57">A863</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E58">A864</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E59">A865</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E60">A866</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E61">A867</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_2_1_1_3_1_1_1_2_1_1"/>
<font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_2"><i><font color="Green" title="E37">A838</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1_1_3"><i><font color="Green" title="E49">A851</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E55">A861</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E56">A862</font></i></a>, <a class="ref" href="gobrd13.html#T47" title="GOBRD13:th.47">GOBRD13:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3_1_1_1_2_1_2"/>
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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#K5" title="GOBRD13:func.5">front_right_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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> )
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_3_1_1_1_2_1_2"/><b>then </b>
( <font color="Maroon" title="c3">f1</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c7">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; <font color="Maroon" title="c4">f2</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c7">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="jordan9.html#E7:33_1_2"><i><font color="Green" title="E36">A837</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_2"><i><font color="Green" title="E42">A843</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E52">A854</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E55">A861</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E56">A862</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E57">A863</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E58">A864</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E59">A865</font></i></a>, <a class="txt" href="jordan9.html#E10:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E60">A866</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E61">A867</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_2_1_1_3_1_1_1_2_1_2"/>
<font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_2"><i><font color="Green" title="E37">A838</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1_1_3"><i><font color="Green" title="E49">A851</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E55">A861</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E56">A862</font></i></a>, <a class="ref" href="gobrd13.html#T49" title="GOBRD13:th.49">GOBRD13:49</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3_1_1_1_2_1_3"/>
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <font color="Maroon" title="c3">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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>
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_3_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon" title="c3">f1</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c7">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; <font color="Maroon" title="c4">f2</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c7">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="jordan9.html#E7:33_1_2"><i><font color="Green" title="E36">A837</font></i></a>, <a class="txt" href="jordan9.html#E14:33_1_2"><i><font color="Green" title="E42">A843</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E52">A854</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E55">A861</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E56">A862</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E57">A863</font></i></a>, <a class="txt" href="jordan9.html#E9:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E59">A865</font></i></a>, <a class="txt" href="jordan9.html#E11:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E61">A867</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_2_1_1_3_1_1_1_2_1_3"/>
<font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E8:33_1_2"><i><font color="Green" title="E37">A838</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1_1_3"><i><font color="Green" title="E49">A851</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E55">A861</font></i></a>, <a class="txt" href="jordan9.html#E6:33_1_2_1_1_3_1_1_1"><i><font color="Green" title="E56">A862</font></i></a>, <a class="ref" href="gobrd13.html#T48" title="GOBRD13:th.48">GOBRD13:48</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:33_1_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_3_1_1_2"/><i><font color="Green" title="E51">A868</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font>
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E52">A869</font></i>: 
<font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/><a NAME="E3:33_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E53">A870</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E4:33_1_2_1_1_3_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</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="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_2"><i><font color="Green" title="E52">A869</font></i></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E5:33_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E54">A871</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/><a NAME="E6:33_1_2_1_1_3_1_1_2"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</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="c4">f2</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="txt" href="jordan9.html#E4:33_1_2_1"><i><font color="Green" title="E48">A849</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_2"><i><font color="Green" title="E52">A869</font></i></a>, <a class="ref" href="finseq_5.html#T18" title="FINSEQ_5:th.18">FINSEQ_5:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:33_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E55">A872</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</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="c3">f1</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3"><i><font color="Green" title="E50">A852</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_2"><i><font color="Green" title="E52">A869</font></i></a>, <a class="txt" href="jordan9.html#E3:33_1_2_1_1_3_1_1_2"><i><font color="Green" title="E53">A870</font></i></a>, <a class="txt" href="jordan9.html#E5:33_1_2_1_1_3_1_1_2"><i><font color="Green" title="E54">A871</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a>, <a class="ref" href="goboard7.html#T40" title="GOBOARD7:th.40">GOBOARD7:40</a></i>;<br/><a NAME="E8:33_1_2_1_1_3_1_1_2"/><i><font color="Green" title="E56">A873</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> <font color="Maroon" title="c3">f1</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2_1_1_3_1_1_2"><i><font color="Green" title="E51">A868</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon" title="c3">f1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon" title="c4">f2</font>
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_2_1"><i><font color="Green" title="E45">A846</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_2_1_1_3_1_1_2"><i><font color="Green" title="E51">A868</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_2_1_1_3_1_1_2"><i><font color="Green" title="E52">A869</font></i></a>, <a class="txt" href="jordan9.html#E7:33_1_2_1_1_3_1_1_2"><i><font color="Green" title="E55">A872</font></i></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E7:33_1_2_1_1_3_1_1_2"><i><font color="Green" title="E55">A872</font></i></a>, <a class="txt" href="jordan9.html#E8:33_1_2_1_1_3_1_1_2"><i><font color="Green" title="E56">A873</font></i></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_2_1_1_3"/>
<font color="Maroon" title="c3">f1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E17:33_1_2"/>
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>]
 
<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="jordan9.html#E15:33_1_2"><i><font color="Green" title="E43">A844</font></i></a>, <a class="txt" href="jordan9.html#E16:33_1_2"><i><font color="Green" title="E44">A845</font></i></a>);<br/></i>
<b>hence </b><a NAME="E18:33_1_2"/>
<font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self" title="JORDAN9:th.4">Th4</a></i>;<br/>


</div>
