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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green">E52</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font>
 ;<br/>

<b>set </b><font color="Maroon">F</font> =  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font>;<br/>
<a NAME="E3:42"/><i><font color="Green">E59</font></i>: 
 <a href="gobrd13.html#K2" target="_self">Values</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K2" target="_self">Values</a> <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:42"/>
<font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="goboard5.html#D5">GOBOARD5:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:42"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E7:42"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E8:42"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E9:42"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E10:42"/>
( ( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> ) or ( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> ) or ( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> ) or ( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E11:42"/><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E12:42"/><i><font color="Green">E65</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E13:42"/><b>then </b><i><font color="Green">E66</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/>
<a NAME="E14:42"/><i><font color="Green">E67</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/>
<b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:42"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E17:42"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E18:42"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E19:42"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 <b>and </b><br/><a NAME="E20:42"/><i><font color="Green">E80</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E21:42"/><i><font color="Green">E94</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E22:42"/><i><font color="Green">E95</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E23:42"/><b>then </b><i><font color="Green">E96</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/>
<a NAME="E24:42"/><i><font color="Green">E97</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/>
<a NAME="E25:42"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E26:42"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E27:42"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><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">E101</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<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">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:42_1_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , <a class="ref" href="sprect_3.html#T24">SPRECT_3:24</a></i>;<br/><a NAME="E3:42_1_1_1"/>
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:42_1_1_1"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div><a NAME="E3:42_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">E104</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:42_1_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:42_1_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:42_1_1_2"/><i><font color="Green">E107</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i'</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i'</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan5d.html#T10">JORDAN5D:10</a></i>;<br/><a NAME="E7:42_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i'</font> &amp; <font color="Maroon">i'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E8:42_1_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i'</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><b>consider </b><font color="Maroon">i1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:42_1_1_2"/><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E11:42_1_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E3:42"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E12:42_1_1_2"/><i><font color="Green">E112</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">i1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">j1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E63</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E13:42_1_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E14:42_1_1_2"/><i><font color="Green">E114</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E65</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><div><i><font color="Green">E115</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">j1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j1'</font>
 ;<br/><a NAME="E2:42_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E65</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green">E66</font></i></a>, <a class="ref" href="sprect_3.html#T24">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#E2:42"><i><font color="Green">E58</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E64</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<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">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1'</font>
 ;<br/><a NAME="E2:42_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E65</font></i></a>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E67</font></i></a>, <a class="ref" href="sprect_3.html#T24">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#E1:42"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E64</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<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#E16:42"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><a NAME="E5:42_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:42_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
(  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp;  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font> )
 <b>by </b><i>, , , , , , , , , , , <a class="ref" href="gobrd13.html#T3" target="_self">Th3</a>, , , <a class="ref" href="goboard5.html#D6">GOBOARD5:def 6</a></i>;<br/><a NAME="E7:42_1_1"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">k</font> )
 <b>by </b><i>, , , , , , , , , , , <a class="ref" href="gobrd13.html#T3" target="_self">Th3</a>, , , , <a class="ref" href="goboard5.html#D7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E8:42_1_1"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> &amp;  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> )
 <b>by </b><i>, , , , , <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E58</font></i></a>, , </i>;<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">E118</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<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">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:42_1_2_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/><a NAME="E3:42_1_2_1"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:42_1_2_1"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><a NAME="E3:42_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">E122</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E2:42_1_2_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:42_1_2_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:42_1_2_2"/><i><font color="Green">E125</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_2_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j'</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan5d.html#T9">JORDAN5D:9</a></i>;<br/><a NAME="E7:42_1_2_2"/><i><font color="Green">E127</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:42_1_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j'</font> &amp; <font color="Maroon">j'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E9:42_1_2_2"/><b>then </b><i><font color="Green">E128</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j'</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E10:42_1_2_2"/><i><font color="Green">E129</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , <a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><b>consider </b><font color="Maroon">i1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:42_1_2_2"/><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E13:42_1_2_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E3:42"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E14:42_1_2_2"/><i><font color="Green">E132</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">i1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">j1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E65</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E15:42_1_2_2"/><b>then </b><i><font color="Green">E133</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E16:42_1_2_2"/><i><font color="Green">E134</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><div><i><font color="Green">E135</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">i1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i1'</font>
 ;<br/><a NAME="E2:42_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sprect_3.html#T25">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#E2:42"><i><font color="Green">E58</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<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">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1'</font>
 ;<br/><a NAME="E2:42_1_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E69</font></i></a>, <a class="ref" href="sprect_3.html#T25">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#E7:42"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E64</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:42_1_2_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><a NAME="E5:42_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:42_1_2"/><b>then </b><i><font color="Green">E137</font></i>: 
(  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, , , , , , , , , , , <a class="ref" href="gobrd13.html#T3" target="_self">Th3</a>, , , <a class="ref" href="goboard5.html#D6">GOBOARD5:def 6</a></i>;<br/><a NAME="E7:42_1_2"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">k</font> )
 <b>by </b><i>, , , , , , , , , , , <a class="ref" href="gobrd13.html#T3" target="_self">Th3</a>, , , , <a class="ref" href="goboard5.html#D7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E8:42_1_2"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> &amp;  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> )
 <b>by </b><i>, , , , , <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E58</font></i></a>, , </i>;<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">E138</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<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">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E2:42_1_3_1"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="ref" href="sprect_3.html#T25">SPRECT_3:25</a></i>;<br/><a NAME="E3:42_1_3_1"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:42_1_3_1"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><a NAME="E3:42_1_3"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">E141</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:42_1_3_2"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:42_1_3_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:42_1_3_2"/><i><font color="Green">E144</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_3_2"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j'</font>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan5d.html#T9">JORDAN5D:9</a></i>;<br/><a NAME="E7:42_1_3_2"/><i><font color="Green">E146</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E8:42_1_3_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j'</font> &amp; <font color="Maroon">j'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E9:42_1_3_2"/><b>then </b><i><font color="Green">E147</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j'</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E10:42_1_3_2"/><i><font color="Green">E148</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, , <a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><b>consider </b><font color="Maroon">i1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:42_1_3_2"/><i><font color="Green">E149</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E13:42_1_3_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E3:42"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E14:42_1_3_2"/><i><font color="Green">E151</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">i1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">j1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E65</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E15:42_1_3_2"/><b>then </b><i><font color="Green">E175</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><a NAME="E16:42_1_3_2"/><i><font color="Green">E176</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard5.html#T3">GOBOARD5:3</a></i>;<br/><div><i><font color="Green">E177</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">i2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i1'</font>
 ;<br/><a NAME="E2:42_1_3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E69</font></i></a>, <a class="ref" href="sprect_3.html#T25">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#E2:42"><i><font color="Green">E58</font></i></a>, <a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E61</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<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">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1'</font>
 ;<br/><a NAME="E2:42_1_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sprect_3.html#T25">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">E61</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E64</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:42_1_3_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><a NAME="E5:42_1_3"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:42_1_3"/><b>then </b><i><font color="Green">E179</font></i>: 
(  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> &amp;  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, , , , , , , , , , , <a class="ref" href="gobrd13.html#T3" target="_self">Th3</a>, , , <a class="ref" href="goboard5.html#D6">GOBOARD5:def 6</a></i>;<br/><a NAME="E7:42_1_3"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, , , , , , , , , , , <a class="ref" href="gobrd13.html#T3" target="_self">Th3</a>, , , , <a class="ref" href="goboard5.html#D7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E8:42_1_3"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> &amp;  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> )
 <b>by </b><i>, , , , , <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E58</font></i></a>, , </i>;<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">E180</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><div><i><font color="Green">E181</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">E182</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<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">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">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">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_4_1_1_1"/>
contradiction
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#T1" target="_self">Th1</a>, , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<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">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_4_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#T1" target="_self">Th1</a>, , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<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">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:42_1_4_2"/><b>then </b><i><font color="Green">E183</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">k</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , , <a class="ref" href="sprect_3.html#T24">SPRECT_3:24</a></i>;<br/><a NAME="E3:42_1_4_2"/>
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:42_1_4_2"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div><a NAME="E4:42_1_4"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">E185</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:42_1_4_3"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:42_1_4_3"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:42_1_4_3"/><i><font color="Green">E188</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i'</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E6:42_1_4_3"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i'</font>,<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan5d.html#T10">JORDAN5D:10</a></i>;<br/><a NAME="E7:42_1_4_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i'</font> &amp; <font color="Maroon">i'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E7:42"><i><font color="Green">E61</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E8:42_1_4_3"/><b>then </b><i><font color="Green">E190</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i'</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E3:42"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E9:42_1_4_3"/><i><font color="Green">E191</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , <a class="txt" href="gobrd13.html#E1:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E3:42"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><b>consider </b><font color="Maroon">i1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:42_1_4_3"/><i><font color="Green">E192</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E12:42_1_4_3"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E6:42"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E13:42_1_4_3"/><i><font color="Green">E194</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">i1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">j1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="txt" href="gobrd13.html#E12:42"><i><font color="Green">E65</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E14:42_1_4_3"/><b>then </b><i><font color="Green">E195</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E15:42_1_4_3"/><i><font color="Green">E196</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><div><i><font color="Green">E197</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">j2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j1'</font>
 ;<br/><a NAME="E2:42_1_4_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E17:42"><i><font color="Green">E69</font></i></a>, <a class="ref" href="sprect_3.html#T24">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#E3:42"><i><font color="Green">E59</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<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">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1'</font>
 ;<br/><a NAME="E2:42_1_4_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="txt" href="gobrd13.html#E14:42"><i><font color="Green">E67</font></i></a>, <a class="txt" href="gobrd13.html#E16:42"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sprect_3.html#T24">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#E1:42"><i><font color="Green">E52</font></i></a>, <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E58</font></i></a>, <a class="txt" href="gobrd13.html#E8:42"><i><font color="Green">E62</font></i></a>, <a class="txt" href="gobrd13.html#E9:42"><i><font color="Green">E63</font></i></a>, <a class="txt" href="gobrd13.html#E11:42"><i><font color="Green">E64</font></i></a>, <a class="txt" href="gobrd13.html#E13:42"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:42_1_4_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><a NAME="E6:42_1_4"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E7:42_1_4"/><b>then </b><i><font color="Green">E199</font></i>: 
(  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> &amp;  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j</font> )
 <b>by </b><i>, , , , , , , , , , , <a class="ref" href="gobrd13.html#T3" target="_self">Th3</a>, , , <a class="ref" href="goboard5.html#D6">GOBOARD5:def 6</a></i>;<br/><a NAME="E8:42_1_4"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i1</font>,<font color="Maroon">j2</font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, , , , , , , , , , , <a class="ref" href="gobrd13.html#T3" target="_self">Th3</a>, , <a class="txt" href="gobrd13.html#E2:42"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="goboard5.html#D7">GOBOARD5:def 7</a></i>;<br/><b>hence </b><a NAME="E9:42_1_4"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> &amp;  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font> )
 <b>by </b><i>, , , , , , , <a class="txt" href="gobrd13.html#E3:42"><i><font color="Green">E59</font></i></a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
