<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">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:()"><i><font color="Red">::  thesis: </font></i></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><br/> for <font color="Olive" title="b2">f</font> being <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</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="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">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="Olive" title="b2">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">n</font> )</span><br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</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="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="Maroon" title="c2">G</font> holds <br/>(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">n</font> )</span><br/><b>let </b><font color="Maroon" title="c3">f</font> be  <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</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">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="c2">G</font> implies (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> ) )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green" title="E32">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</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">f</font> )
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> )</span><br/>

<b>set </b><font color="Maroon" title="c4">F</font> =  <a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font>;<br/>
<a NAME="E3:42"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="gobrd13.html#T21" target="_self" title="GOBRD13:th.21">Th21</a></i>;<br/>
<a NAME="E4:42"/>
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="ref" href="goboard5.html#D5" title="GOBOARD5:def.5">GOBOARD5:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">m</font>, <font color="Maroon" title="c6">k</font>, <font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:42"/><i><font color="Green" title="E35">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>and </b><br/><a NAME="E7:42"/><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">k</font>
 <b>and </b><br/><a NAME="E8:42"/><i><font color="Green" title="E37">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>and </b><br/><a NAME="E9:42"/><i><font color="Green" title="E38">A7</font></i>: 
<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">n</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>
 <b>and </b><br/><a NAME="E10:42"/>
( ( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c6">k</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="c8">j</font> ) or ( <font color="Maroon" title="c5">m</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="c7">i</font> &amp; <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> ) or ( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/>
<a NAME="E11:42"/><i><font color="Green" title="E39">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E6:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/>
<a NAME="E12:42"/><i><font color="Green" title="E40">A9</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E8:42"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/>
<a NAME="E13:42"/><b>then </b><i><font color="Green" title="E41">A10</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/>
<a NAME="E14:42"/><i><font color="Green" title="E42">A11</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">j1</font>, <font color="Maroon" title="c11">i2</font>, <font color="Maroon" title="c12">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E16:42"/><i><font color="Green" title="E43">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E17:42"/><i><font color="Green" title="E44">A13</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font>
 <b>and </b><br/><a NAME="E18:42"/><i><font color="Green" title="E45">A14</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E19:42"/><i><font color="Green" title="E46">A15</font></i>: 
<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">n</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="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font>
 <b>and </b><br/><a NAME="E20:42"/><i><font color="Green" title="E47">A16</font></i>: 
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/>
<a NAME="E21:42"/><i><font color="Green" title="E48">A17</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i1</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j1</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E16:42"><i><font color="Green" title="E43">A12</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/>
<a NAME="E22:42"/><i><font color="Green" title="E49">A18</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c11">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j2</font> &amp; <font color="Maroon" title="c12">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E18:42"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/>
<a NAME="E23:42"/><b>then </b><i><font color="Green" title="E50">A19</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/>
<a NAME="E24:42"/><i><font color="Green" title="E51">A20</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/>
<a NAME="E25:42"/><i><font color="Green" title="E52">A21</font></i>: 
( <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E26:42"/><i><font color="Green" title="E53">A22</font></i>: 
( <font color="Maroon" title="c6">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> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c8">j</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="c8">j</font> &amp; <font color="Maroon" title="c5">m</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="c5">m</font> &amp; <font color="Maroon" title="c7">i</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="c7">i</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="E27:42"/><i><font color="Green" title="E54">A23</font></i>: 
( <font color="Maroon" title="c9">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="c9">i1</font> &amp; <font color="Maroon" title="c10">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="c10">j1</font> &amp; <font color="Maroon" title="c11">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="c11">i2</font> &amp; <font color="Maroon" title="c12">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="c12">j2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="gobrd13.html#E20:42"><i><font color="Green" title="E47">A16</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>suppose </b></a><a NAME="E1:42_1_1"/><i><font color="Green" title="E55">A24</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_1"/>
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_1_1"/><b>then </b><i><font color="Green" title="E56">A25</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green" title="E42">A11</font></i></a>, <a class="ref" href="sprect_3.html#T24" title="SPRECT_3:th.24">SPRECT_3:24</a></i>;<br/><a NAME="E3:42_1_1_1"/>
<font color="Maroon" title="c10">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j2</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_1"><i><font color="Green" title="E55">A24</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><a NAME="E4:42_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E24:42"><i><font color="Green" title="E51">A20</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_1_1"><i><font color="Green" title="E56">A25</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E3:42_1_1"/><b>then </b><i><font color="Green" title="E56">A26</font></i>: 
<font color="Maroon" title="c6">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> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2"/><i><font color="Green" title="E57">A27</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_1_2"/><b>then </b><i><font color="Green" title="E58">A28</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">l</font>, <font color="Maroon" title="c14">i'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:42_1_1_2"/><i><font color="Green" title="E59">A29</font></i>: 
<font color="Maroon" title="c13">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E5:42_1_1_2"/><i><font color="Green" title="E60">A30</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_1_2"/><i><font color="Green" title="E61">A31</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">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="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="jordan5d.html#T10" title="JORDAN5D:th.10">JORDAN5D:10</a></i>;<br/><a NAME="E7:42_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i'</font> &amp; <font color="Maroon" title="c14">i'</font> <a href="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="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E5:42_1_1_2"><i><font color="Green" title="E60">A30</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/><a NAME="E8:42_1_1_2"/><b>then </b><i><font color="Green" title="E62">A32</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_1_2"><i><font color="Green" title="E58">A28</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">i1'</font>, <font color="Maroon" title="c16">j1'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:42_1_1_2"/><i><font color="Green" title="E63">A33</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E11:42_1_1_2"/><i><font color="Green" title="E64">A34</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E4:42_1_1_2"><i><font color="Green" title="E59">A29</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><a NAME="E12:42_1_1_2"/><i><font color="Green" title="E65">A35</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i1'</font> &amp; <font color="Maroon" title="c15">i1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1'</font> &amp; <font color="Maroon" title="c16">j1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E10:42_1_1_2"><i><font color="Green" title="E63">A33</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/><a NAME="E13:42_1_1_2"/><b>then </b><i><font color="Green" title="E66">A36</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/><a NAME="E14:42_1_1_2"/><i><font color="Green" title="E67">A37</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E12:42_1_1_2"><i><font color="Green" title="E65">A35</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/><div><i><font color="Green" title="E68">A38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2_1"/>
<font color="Maroon" title="c10">j1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c16">j1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="gobrd13.html#E24:42"><i><font color="Green" title="E51">A20</font></i></a>, <a class="txt" href="gobrd13.html#E12:42_1_1_2"><i><font color="Green" title="E65">A35</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_1_2"><i><font color="Green" title="E66">A36</font></i></a>, <a class="ref" href="sprect_3.html#T24" title="SPRECT_3:th.24">SPRECT_3:24</a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_1_2"><i><font color="Green" title="E58">A28</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_1_2"><i><font color="Green" title="E61">A31</font></i></a>, <a class="txt" href="gobrd13.html#E8:42_1_1_2"><i><font color="Green" title="E62">A32</font></i></a>, <a class="txt" href="gobrd13.html#E11:42_1_1_2"><i><font color="Green" title="E64">A34</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_2_2"/>
<font color="Maroon" title="c12">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E12:42_1_1_2"><i><font color="Green" title="E65">A35</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_1_2"><i><font color="Green" title="E67">A37</font></i></a>, <a class="ref" href="sprect_3.html#T24" title="SPRECT_3:th.24">SPRECT_3:24</a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_1_2"><i><font color="Green" title="E57">A27</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_1_2"><i><font color="Green" title="E61">A31</font></i></a>, <a class="txt" href="gobrd13.html#E8:42_1_1_2"><i><font color="Green" title="E62">A32</font></i></a>, <a class="txt" href="gobrd13.html#E11:42_1_1_2"><i><font color="Green" title="E64">A34</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E17:42_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_1"><i><font color="Green" title="E55">A24</font></i></a>, <a class="txt" href="gobrd13.html#E15:42_1_1_2"><i><font color="Green" title="E68">A38</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:42_1_1"/><b>then </b><i><font color="Green" title="E57">A39</font></i>: 
<font color="Maroon" title="c6">k</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="c8">j</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42_1_1"><i><font color="Green" title="E56">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:42_1_1"/><b>then </b><i><font color="Green" title="E58">A40</font></i>: 
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">k</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_1"><i><font color="Green" title="E55">A24</font></i></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a>, <a class="ref" href="goboard5.html#D6" title="GOBOARD5:def.6">GOBOARD5:def 6</a></i>;<br/><a NAME="E7:42_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j1</font> &amp;  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c6">k</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_1"><i><font color="Green" title="E55">A24</font></i></a>, <a class="txt" href="gobrd13.html#E5:42_1_1"><i><font color="Green" title="E57">A39</font></i></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a>, <a class="ref" href="goboard5.html#D7" title="GOBOARD5:def.7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E8:42_1_1"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_1"><i><font color="Green" title="E58">A40</font></i></a>, <a class="ref" href="gobrd13.html#T18" target="_self" title="GOBRD13:th.18">Th18</a>, <a class="ref" href="gobrd13.html#T19" target="_self" title="GOBRD13:th.19">Th19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>suppose </b></a><a NAME="E1:42_1_2"/><i><font color="Green" title="E55">A41</font></i>: 
( <font color="Maroon" title="c9">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="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_2_1"/>
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_2_1"/><b>then </b><i><font color="Green" title="E56">A42</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="sprect_3.html#T25" title="SPRECT_3:th.25">SPRECT_3:25</a></i>;<br/><a NAME="E3:42_1_2_1"/>
<font color="Maroon" title="c9">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i2</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_2"><i><font color="Green" title="E55">A41</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><a NAME="E4:42_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_2"><i><font color="Green" title="E55">A41</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_2_1"><i><font color="Green" title="E56">A42</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E3:42_1_2"/><b>then </b><i><font color="Green" title="E56">A43</font></i>: 
<font color="Maroon" title="c5">m</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> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_2_2"/><i><font color="Green" title="E57">A44</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_2_2"/><b>then </b><i><font color="Green" title="E58">A45</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">l</font>, <font color="Maroon" title="c14">j'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:42_1_2_2"/><i><font color="Green" title="E59">A46</font></i>: 
<font color="Maroon" title="c13">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E5:42_1_2_2"/><i><font color="Green" title="E60">A47</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_2_2"/><i><font color="Green" title="E61">A48</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</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">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j'</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="jordan5d.html#T9" title="JORDAN5D:th.9">JORDAN5D:9</a></i>;<br/><a NAME="E7:42_1_2_2"/><i><font color="Green" title="E62">A49</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E8:42_1_2_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j'</font> &amp; <font color="Maroon" title="c14">j'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E5:42_1_2_2"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/><a NAME="E9:42_1_2_2"/><b>then </b><i><font color="Green" title="E63">A50</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j'</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_2_2"><i><font color="Green" title="E58">A45</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/><a NAME="E10:42_1_2_2"/><i><font color="Green" title="E64">A51</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_2_2"><i><font color="Green" title="E58">A45</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">i1'</font>, <font color="Maroon" title="c16">j1'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:42_1_2_2"/><i><font color="Green" title="E65">A52</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E13:42_1_2_2"/><i><font color="Green" title="E66">A53</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E4:42_1_2_2"><i><font color="Green" title="E59">A46</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><a NAME="E14:42_1_2_2"/><i><font color="Green" title="E67">A54</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i1'</font> &amp; <font color="Maroon" title="c15">i1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1'</font> &amp; <font color="Maroon" title="c16">j1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42_1_2_2"><i><font color="Green" title="E65">A52</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/><a NAME="E15:42_1_2_2"/><b>then </b><i><font color="Green" title="E68">A55</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/><a NAME="E16:42_1_2_2"/><i><font color="Green" title="E69">A56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_2_2"><i><font color="Green" title="E67">A54</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/><div><i><font color="Green" title="E70">A57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_2_2_1"/>
<font color="Maroon" title="c9">i1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_2_2"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="gobrd13.html#E15:42_1_2_2"><i><font color="Green" title="E68">A55</font></i></a>, <a class="ref" href="sprect_3.html#T25" title="SPRECT_3:th.25">SPRECT_3:25</a></i>;<br/><b>hence </b><a NAME="E3:42_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_2_2"><i><font color="Green" title="E58">A45</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_2_2"><i><font color="Green" title="E61">A48</font></i></a>, <a class="txt" href="gobrd13.html#E9:42_1_2_2"><i><font color="Green" title="E63">A50</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_2_2"><i><font color="Green" title="E66">A53</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_2_2_2"/>
<font color="Maroon" title="c11">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_2_2"><i><font color="Green" title="E67">A54</font></i></a>, <a class="txt" href="gobrd13.html#E16:42_1_2_2"><i><font color="Green" title="E69">A56</font></i></a>, <a class="ref" href="sprect_3.html#T25" title="SPRECT_3:th.25">SPRECT_3:25</a></i>;<br/><b>hence </b><a NAME="E3:42_1_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_2_2"><i><font color="Green" title="E57">A44</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_2_2"><i><font color="Green" title="E61">A48</font></i></a>, <a class="txt" href="gobrd13.html#E7:42_1_2_2"><i><font color="Green" title="E62">A49</font></i></a>, <a class="txt" href="gobrd13.html#E9:42_1_2_2"><i><font color="Green" title="E63">A50</font></i></a>, <a class="txt" href="gobrd13.html#E10:42_1_2_2"><i><font color="Green" title="E64">A51</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_2_2"><i><font color="Green" title="E66">A53</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E19:42_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_2"><i><font color="Green" title="E55">A41</font></i></a>, <a class="txt" href="gobrd13.html#E17:42_1_2_2"><i><font color="Green" title="E70">A57</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:42_1_2"/><b>then </b><i><font color="Green" title="E57">A58</font></i>: 
<font color="Maroon" title="c5">m</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="c7">i</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42_1_2"><i><font color="Green" title="E56">A43</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:42_1_2"/><b>then </b><i><font color="Green" title="E58">A59</font></i>: 
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c9">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c5">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_2"><i><font color="Green" title="E55">A41</font></i></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a>, <a class="ref" href="goboard5.html#D6" title="GOBOARD5:def.6">GOBOARD5:def 6</a></i>;<br/><a NAME="E7:42_1_2"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font> &amp;  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">k</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_2"><i><font color="Green" title="E55">A41</font></i></a>, <a class="txt" href="gobrd13.html#E5:42_1_2"><i><font color="Green" title="E57">A58</font></i></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a>, <a class="ref" href="goboard5.html#D7" title="GOBOARD5:def.7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E8:42_1_2"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_2"><i><font color="Green" title="E58">A59</font></i></a>, <a class="ref" href="gobrd13.html#T18" target="_self" title="GOBRD13:th.18">Th18</a>, <a class="ref" href="gobrd13.html#T20" target="_self" title="GOBRD13:th.20">Th20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_3"><b>suppose </b></a><a NAME="E1:42_1_3"/><i><font color="Green" title="E55">A60</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_1"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_3_1"/><b>then </b><i><font color="Green" title="E56">A61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="sprect_3.html#T25" title="SPRECT_3:th.25">SPRECT_3:25</a></i>;<br/><a NAME="E3:42_1_3_1"/>
<font color="Maroon" title="c9">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">i2</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_3"><i><font color="Green" title="E55">A60</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><a NAME="E4:42_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_3"><i><font color="Green" title="E55">A60</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_3_1"><i><font color="Green" title="E56">A61</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E3:42_1_3"/><b>then </b><i><font color="Green" title="E56">A62</font></i>: 
<font color="Maroon" title="c7">i</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> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_2"/><i><font color="Green" title="E57">A63</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_3_2"/><b>then </b><i><font color="Green" title="E58">A64</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">l</font>, <font color="Maroon" title="c14">j'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:42_1_3_2"/><i><font color="Green" title="E59">A65</font></i>: 
<font color="Maroon" title="c13">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E5:42_1_3_2"/><i><font color="Green" title="E60">A66</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_3_2"/><i><font color="Green" title="E61">A67</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</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="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j'</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="jordan5d.html#T9" title="JORDAN5D:th.9">JORDAN5D:9</a></i>;<br/><a NAME="E7:42_1_3_2"/><i><font color="Green" title="E62">A68</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E8:42_1_3_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">j'</font> &amp; <font color="Maroon" title="c14">j'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E5:42_1_3_2"><i><font color="Green" title="E60">A66</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/><a NAME="E9:42_1_3_2"/><b>then </b><i><font color="Green" title="E63">A69</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">j'</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_3_2"><i><font color="Green" title="E58">A64</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/><a NAME="E10:42_1_3_2"/><i><font color="Green" title="E64">A70</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_3_2"><i><font color="Green" title="E58">A64</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">i1'</font>, <font color="Maroon" title="c16">j1'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:42_1_3_2"/><i><font color="Green" title="E65">A71</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E13:42_1_3_2"/><i><font color="Green" title="E66">A72</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E4:42_1_3_2"><i><font color="Green" title="E59">A65</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><a NAME="E14:42_1_3_2"/><i><font color="Green" title="E67">A73</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i1'</font> &amp; <font color="Maroon" title="c15">i1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1'</font> &amp; <font color="Maroon" title="c16">j1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42_1_3_2"><i><font color="Green" title="E65">A71</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/><a NAME="E15:42_1_3_2"/><b>then </b><i><font color="Green" title="E68">A74</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/><a NAME="E16:42_1_3_2"/><i><font color="Green" title="E69">A75</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_3_2"><i><font color="Green" title="E67">A73</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<br/><div><i><font color="Green" title="E70">A76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_2_1"/>
<font color="Maroon" title="c11">i2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_3_2"><i><font color="Green" title="E67">A73</font></i></a>, <a class="txt" href="gobrd13.html#E16:42_1_3_2"><i><font color="Green" title="E69">A75</font></i></a>, <a class="ref" href="sprect_3.html#T25" title="SPRECT_3:th.25">SPRECT_3:25</a></i>;<br/><b>hence </b><a NAME="E3:42_1_3_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_3_2"><i><font color="Green" title="E58">A64</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_3_2"><i><font color="Green" title="E61">A67</font></i></a>, <a class="txt" href="gobrd13.html#E9:42_1_3_2"><i><font color="Green" title="E63">A69</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_3_2"><i><font color="Green" title="E66">A72</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_3_2_2"/>
<font color="Maroon" title="c9">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_3_2"><i><font color="Green" title="E67">A73</font></i></a>, <a class="txt" href="gobrd13.html#E15:42_1_3_2"><i><font color="Green" title="E68">A74</font></i></a>, <a class="ref" href="sprect_3.html#T25" title="SPRECT_3:th.25">SPRECT_3:25</a></i>;<br/><b>hence </b><a NAME="E3:42_1_3_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_3_2"><i><font color="Green" title="E57">A63</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_3_2"><i><font color="Green" title="E61">A67</font></i></a>, <a class="txt" href="gobrd13.html#E7:42_1_3_2"><i><font color="Green" title="E62">A68</font></i></a>, <a class="txt" href="gobrd13.html#E9:42_1_3_2"><i><font color="Green" title="E63">A69</font></i></a>, <a class="txt" href="gobrd13.html#E10:42_1_3_2"><i><font color="Green" title="E64">A70</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_3_2"><i><font color="Green" title="E66">A72</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E19:42_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_3"><i><font color="Green" title="E55">A60</font></i></a>, <a class="txt" href="gobrd13.html#E17:42_1_3_2"><i><font color="Green" title="E70">A76</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:42_1_3"/><b>then </b><i><font color="Green" title="E57">A77</font></i>: 
<font color="Maroon" title="c7">i</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="c5">m</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42_1_3"><i><font color="Green" title="E56">A62</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:42_1_3"/><b>then </b><i><font color="Green" title="E58">A78</font></i>: 
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_3"><i><font color="Green" title="E55">A60</font></i></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a>, <a class="ref" href="goboard5.html#D6" title="GOBOARD5:def.6">GOBOARD5:def 6</a></i>;<br/><a NAME="E7:42_1_3"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c7">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_3"><i><font color="Green" title="E55">A60</font></i></a>, <a class="txt" href="gobrd13.html#E5:42_1_3"><i><font color="Green" title="E57">A77</font></i></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a>, <a class="ref" href="goboard5.html#D7" title="GOBOARD5:def.7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E8:42_1_3"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_3"><i><font color="Green" title="E58">A78</font></i></a>, <a class="ref" href="gobrd13.html#T18" target="_self" title="GOBRD13:th.18">Th18</a>, <a class="ref" href="gobrd13.html#T20" target="_self" title="GOBRD13:th.20">Th20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4"><b>suppose </b></a><a NAME="E1:42_1_4"/><i><font color="Green" title="E55">A79</font></i>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> )</span><br/><div class="add"><div><i><font color="Green" title="E56">A80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_4_1"/><i><font color="Green" title="E57">A81</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_4_1_1"/>
( <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">i</font> or <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">i</font> )
 </span><b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_4_1"><i><font color="Green" title="E57">A81</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_4_1_1_1"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:42_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E23:42"><i><font color="Green" title="E50">A19</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green" title="E55">A79</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_4_1_1_2"/>
<font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:42_1_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green" title="E41">A10</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E23:42"><i><font color="Green" title="E50">A19</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green" title="E55">A79</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_4_2"/>
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_4_2"/><b>then </b><i><font color="Green" title="E57">A82</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green" title="E42">A11</font></i></a>, <a class="ref" href="sprect_3.html#T24" title="SPRECT_3:th.24">SPRECT_3:24</a></i>;<br/><a NAME="E3:42_1_4_2"/>
<font color="Maroon" title="c10">j1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">j2</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green" title="E55">A79</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><a NAME="E4:42_1_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E24:42"><i><font color="Green" title="E51">A20</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_4_2"><i><font color="Green" title="E57">A82</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E4:42_1_4"/><b>then </b><i><font color="Green" title="E57">A83</font></i>: 
<font color="Maroon" title="c8">j</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> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_4_3"/><i><font color="Green" title="E58">A84</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_4_3"/><b>then </b><i><font color="Green" title="E59">A85</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">l</font>, <font color="Maroon" title="c14">i'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:42_1_4_3"/><i><font color="Green" title="E60">A86</font></i>: 
<font color="Maroon" title="c13">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 <b>and </b><br/><a NAME="E5:42_1_4_3"/><i><font color="Green" title="E61">A87</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_4_3"/><i><font color="Green" title="E62">A88</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="jordan5d.html#T10" title="JORDAN5D:th.10">JORDAN5D:10</a></i>;<br/><a NAME="E7:42_1_4_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i'</font> &amp; <font color="Maroon" title="c14">i'</font> <a href="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="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E5:42_1_4_3"><i><font color="Green" title="E61">A87</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/><a NAME="E8:42_1_4_3"/><b>then </b><i><font color="Green" title="E63">A89</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_4_3"><i><font color="Green" title="E59">A85</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/><a NAME="E9:42_1_4_3"/><i><font color="Green" title="E64">A90</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">m</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_4_3"><i><font color="Green" title="E59">A85</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">i1'</font>, <font color="Maroon" title="c16">j1'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:42_1_4_3"/><i><font color="Green" title="E65">A91</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E12:42_1_4_3"/><i><font color="Green" title="E66">A92</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E4:42_1_4_3"><i><font color="Green" title="E60">A86</font></i></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></i>;<br/><a NAME="E13:42_1_4_3"/><i><font color="Green" title="E67">A93</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i1'</font> &amp; <font color="Maroon" title="c15">i1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1'</font> &amp; <font color="Maroon" title="c16">j1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E11:42_1_4_3"><i><font color="Green" title="E65">A91</font></i></a>, <a class="ref" href="matrix_1.html#T39" title="MATRIX_1:th.39">MATRIX_1:39</a></i>;<br/><a NAME="E14:42_1_4_3"/><b>then </b><i><font color="Green" title="E68">A94</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/><a NAME="E15:42_1_4_3"/><i><font color="Green" title="E69">A95</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_4_3"><i><font color="Green" title="E67">A93</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>;<br/><div><i><font color="Green" title="E70">A96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_4_3_1"/>
<font color="Maroon" title="c12">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c16">j1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_4_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E22:42"><i><font color="Green" title="E49">A18</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_4_3"><i><font color="Green" title="E67">A93</font></i></a>, <a class="txt" href="gobrd13.html#E15:42_1_4_3"><i><font color="Green" title="E69">A95</font></i></a>, <a class="ref" href="sprect_3.html#T24" title="SPRECT_3:th.24">SPRECT_3:24</a></i>;<br/><b>hence </b><a NAME="E3:42_1_4_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_4_3"><i><font color="Green" title="E59">A85</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_4_3"><i><font color="Green" title="E62">A88</font></i></a>, <a class="txt" href="gobrd13.html#E8:42_1_4_3"><i><font color="Green" title="E63">A89</font></i></a>, <a class="txt" href="gobrd13.html#E12:42_1_4_3"><i><font color="Green" title="E66">A92</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_4_3_2"/>
<font color="Maroon" title="c10">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_1_4_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">i1'</font>,<font color="Maroon" title="c16">j1'</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="gobrd13.html#E21:42"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="gobrd13.html#E24:42"><i><font color="Green" title="E51">A20</font></i></a>, <a class="txt" href="gobrd13.html#E13:42_1_4_3"><i><font color="Green" title="E67">A93</font></i></a>, <a class="txt" href="gobrd13.html#E14:42_1_4_3"><i><font color="Green" title="E68">A94</font></i></a>, <a class="ref" href="sprect_3.html#T24" title="SPRECT_3:th.24">SPRECT_3:24</a></i>;<br/><b>hence </b><a NAME="E3:42_1_4_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E25:42"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4_3"><i><font color="Green" title="E58">A84</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_4_3"><i><font color="Green" title="E62">A88</font></i></a>, <a class="txt" href="gobrd13.html#E8:42_1_4_3"><i><font color="Green" title="E63">A89</font></i></a>, <a class="txt" href="gobrd13.html#E9:42_1_4_3"><i><font color="Green" title="E64">A90</font></i></a>, <a class="txt" href="gobrd13.html#E12:42_1_4_3"><i><font color="Green" title="E66">A92</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E18:42_1_4_3"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green" title="E55">A79</font></i></a>, <a class="txt" href="gobrd13.html#E16:42_1_4_3"><i><font color="Green" title="E70">A96</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E6:42_1_4"/><b>then </b><i><font color="Green" title="E58">A97</font></i>: 
<font color="Maroon" title="c8">j</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="c6">k</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E4:42_1_4"><i><font color="Green" title="E57">A83</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E7:42_1_4"/><b>then </b><i><font color="Green" title="E59">A98</font></i>: 
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green" title="E55">A79</font></i></a>, <a class="ref" href="gobrd13.html#D2" target="_self" title="GOBRD13:def.2">Def2</a>, <a class="ref" href="goboard5.html#D6" title="GOBOARD5:def.6">GOBOARD5:def 6</a></i>;<br/><a NAME="E8:42_1_4"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c12">j2</font> &amp;  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E26:42"><i><font color="Green" title="E53">A22</font></i></a>, <a class="txt" href="gobrd13.html#E27:42"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green" title="E55">A79</font></i></a>, <a class="txt" href="gobrd13.html#E6:42_1_4"><i><font color="Green" title="E58">A97</font></i></a>, <a class="ref" href="gobrd13.html#D3" target="_self" title="GOBRD13:def.3">Def3</a>, <a class="ref" href="goboard5.html#D7" title="GOBOARD5:def.7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E9:42_1_4"/>
(  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> &amp;  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E3:42"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="gobrd13.html#E18:42"><i><font color="Green" title="E45">A14</font></i></a>, <a class="txt" href="gobrd13.html#E19:42"><i><font color="Green" title="E46">A15</font></i></a>, <a class="txt" href="gobrd13.html#E1:42_1_4"><i><font color="Green" title="E55">A79</font></i></a>, <a class="txt" href="gobrd13.html#E2:42_1_4"><i><font color="Green" title="E56">A80</font></i></a>, <a class="txt" href="gobrd13.html#E7:42_1_4"><i><font color="Green" title="E59">A98</font></i></a>, <a class="ref" href="gobrd13.html#T18" target="_self" title="GOBRD13:th.18">Th18</a>, <a class="ref" href="gobrd13.html#T19" target="_self" title="GOBRD13:th.19">Th19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
