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<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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><br/> for <font color="Olive" title="b2">G</font> being  <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">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> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <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> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b2">G</font> holds <br/>(  <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"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b2">G</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b2">G</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <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> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b1">G</font> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">G</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Olive" title="b1">G</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> &amp; <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> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c4">G</font> implies (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> ) )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <span class="kw">and </span><br/><a NAME="E2:40"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:40"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<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> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E4:40"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c4">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> )</span><br/>

<a NAME="E5:40"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E6:40"/>
<font color="Maroon" title="c1">k</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">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="seq_4.html#T151" title="SEQ_4:th.151">SEQ_4:151</a></span>;<br/>
<a NAME="E7:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">lf</font> =  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">lfn</font> =  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">rf</font> =  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">rfn</font> =  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>;<br/>
<a NAME="E8:40"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E9:40"/>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="seq_4.html#T151" title="SEQ_4:th.151">SEQ_4:151</a></span>;<br/>
<a NAME="E10:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">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="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></span>;<br/>
<a NAME="E11:40"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c4">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="jordan8.html#T5" title="JORDAN8:th.5">JORDAN8:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E13:40"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
( <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> <font color="Maroon" title="c4">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <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"><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> <font color="Maroon" title="c4">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">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">G</font> <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 class="kw">and </span><br/><a NAME="E14:40"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
( ( <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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>;<br/>
<a NAME="E15:40"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
( <font color="Maroon" title="c11">j1</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> <font color="Maroon" title="c11">j1</font> &amp; <font color="Maroon" title="c13">j2</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> <font color="Maroon" title="c13">j2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E16:40"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
( <font color="Maroon" title="c10">i1</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> <font color="Maroon" title="c10">i1</font> &amp; <font color="Maroon" title="c12">i2</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> <font color="Maroon" title="c12">i2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40"><span class="lab"><font color="Green" title="E42">A11</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_1"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
( <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#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> )</span><br/><div class="add"><span class="kw">hence </span> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E1:40_1_1_1"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font></span><br/><span class="kw">thus </span> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">j1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E1:40_1_1_1"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a></span>
<br/>.= 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E1:40_1_1_1"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_2"/><span class="lab"><font color="Green" title="E45">A15</font></span>: 
( <font color="Maroon" title="c10">i1</font> <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="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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> )</span><br/><div class="add"><span class="kw">hence </span> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">j1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E1:40_1_1_2"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font></span><br/><span class="kw">thus </span> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c10">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E1:40_1_1_2"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a></span>
<br/>.= 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E1:40_1_1_2"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_3"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
( <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#K2" title="NAT_1:func.2">+</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> )</span><br/><div class="add"><span class="kw">hence </span> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c12">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E1:40_1_1_3"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font></span><br/><span class="kw">thus </span> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c13">j2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E1:40_1_1_3"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a></span>
<br/>.= 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E16:40"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E1:40_1_1_3"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_4"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
( <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#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> )</span><br/><div class="add"><span class="kw">hence </span> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c13">j2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E1:40_1_1_4"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font></span><br/><span class="kw">thus </span> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c4">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E1:40_1_1_4"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a></span>
<br/>.= 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E11:40"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E13:40"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E15:40"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E1:40_1_1_4"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E18:40"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c4">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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