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<b>let </b><font color="Maroon">k</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">f</font> be  <a href="goboard5.html#V2" target="_self">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</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> )
 ;<br/>

<a NAME="E2:41"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:41"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="goboard5.html#D2" target="_self">Def2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:41"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="goboard5.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then 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="E6:41"/><i><font color="Green">E43</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</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">i1</font>,<font color="Maroon">j1</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:41"/>
<font color="Maroon">k</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#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:41"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="goboard5.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><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="E10:41"/><i><font color="Green">E46</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">i2</font>,<font color="Maroon">j2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E11:41"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="goboard5.html#T2" target="_self">Th2</a>, <a class="txt" href="goboard5.html#E1:41"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E4:41"><i><font color="Green">E42</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1"/>
( (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</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;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E8:41"><i><font color="Green">E44</font></i></a>, <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>case </b></a><b>that </b><a NAME="E1:41_1_1_1"/><i><font color="Green">E49</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E2:41_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><a NAME="E3:41_1_1_1"/>
( <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/><a NAME="E4:41_1_1_1"/><b>then </b>
( <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1 or  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E10:41"><i><font color="Green">E46</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:41_1_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 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> )
 ;<br/><b>thus </b><a NAME="E6:41_1_1_1"/>
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="txt" href="goboard5.html#E11:41"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><b>case </b></a><b>that </b><a NAME="E1:41_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>and </b><br/><a NAME="E2:41_1_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E3:41_1_1_2"/>
( <font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 ;<br/><a NAME="E4:41_1_1_2"/><b>then </b>
( <font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 1 or  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E11:41"><i><font color="Green">E47</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:41_1_1_2"/>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 or <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/><b>thus </b><a NAME="E6:41_1_1_2"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="goboard5.html#E10:41"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:41"/><i><font color="Green">E53</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 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> <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="goboard5.html#E1:41"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E14:41"/><i><font color="Green">E54</font></i>: 
( 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> <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="goboard5.html#E6:41"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<a NAME="E15:41"/><i><font color="Green">E57</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 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> <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="goboard5.html#E1:41"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E16:41"/><i><font color="Green">E58</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> <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="goboard5.html#E6:41"><i><font color="Green">E43</font></i></a>, </i>;<br/>
<a NAME="E17:41"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="goboard5.html#E12:41"><i><font color="Green">E48</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:41"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E21:41"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E22:41"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">i</font> <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="txt" href="goboard5.html#E12:41"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E23:41"/>
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="goboard5.html#E10:41"><i><font color="Green">E46</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E25:41"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E27:41"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E13:41"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E28:41"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">j</font> <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="txt" href="goboard5.html#E10:41"><i><font color="Green">E46</font></i></a>, <a class="txt" href="goboard5.html#E14:41"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_2"/>
( ( <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="41_2_1"><b>suppose </b></a><a NAME="E1:41_2_1"/><i><font color="Green">E77</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"><a NAME="E2:41_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">j1</font> <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="txt" href="goboard5.html#E11:41"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:41_2_1"/>
(  <a href="goboard5.html#K5" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3" target="_self">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">j1</font> &amp;  <a href="goboard5.html#K4" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3" target="_self">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">i1</font>,<font color="Maroon">j1</font> )
 <b>by </b><i><a class="ref" href="goboard5.html#D2" target="_self">Def2</a>, <a class="txt" href="goboard5.html#E1:41"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="goboard5.html#T4" target="_self">Th4</a>, </i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="goboard5.html#K5" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> = 
 <a href="topreal1.html#K3">LSeg</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">i1</font>,<font color="Maroon">j1</font></span>)</span>,<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">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E10:41"><i><font color="Green">E46</font></i></a>, , , <a class="txt" href="goboard5.html#E15:41"><i><font color="Green">E57</font></i></a>, </i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="goboard5.html#D2" target="_self">Def2</a>, <a class="txt" href="goboard5.html#E1:41"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard5.html#T4" target="_self">Th4</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_2"><b>suppose </b></a><a NAME="E1:41_2_2"/><i><font color="Green">E79</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"><a NAME="E2:41_2_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">i1</font> <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="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:41_2_2"/>
(  <a href="goboard5.html#K5" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3" target="_self">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">i1</font>,<font color="Maroon">j1</font> &amp;  <a href="goboard5.html#K4" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3" target="_self">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">i1</font>,<font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="goboard5.html#D2" target="_self">Def2</a>, <a class="txt" href="goboard5.html#E1:41"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green">E43</font></i></a>, <a class="txt" href="goboard5.html#E14:41"><i><font color="Green">E54</font></i></a>, <a class="ref" href="goboard5.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard5.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="goboard5.html#K5" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> = 
 <a href="topreal1.html#K3">LSeg</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">i1</font>,<font color="Maroon">j1</font></span>)</span>,<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">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E12:41"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard5.html#E14:41"><i><font color="Green">E54</font></i></a>, <a class="ref" href="goboard5.html#T3" target="_self">Th3</a>, <a class="txt" href="goboard5.html#E15:41"><i><font color="Green">E57</font></i></a>, <a class="ref" href="goboard5.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="goboard5.html#D2" target="_self">Def2</a>, <a class="txt" href="goboard5.html#E1:41"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard5.html#T4" target="_self">Th4</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_3"><b>suppose </b></a><a NAME="E1:41_2_3"/><i><font color="Green">E81</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"><a NAME="E2:41_2_3"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">i2</font> <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="txt" href="goboard5.html#E12:41"><i><font color="Green">E48</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:41_2_3"/>
(  <a href="goboard5.html#K5" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3" target="_self">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">i2</font>,<font color="Maroon">j</font> &amp;  <a href="goboard5.html#K4" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3" target="_self">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">i2</font>,<font color="Maroon">j1</font> )
 <b>by </b><i><a class="ref" href="goboard5.html#D2" target="_self">Def2</a>, <a class="txt" href="goboard5.html#E1:41"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green">E43</font></i></a>, <a class="txt" href="goboard5.html#E14:41"><i><font color="Green">E54</font></i></a>, <a class="ref" href="goboard5.html#T4" target="_self">Th4</a>, </i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="goboard5.html#K5" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> = 
 <a href="topreal1.html#K3">LSeg</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">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j1</font></span>)</span>,<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">i2</font>,<font color="Maroon">j1</font></span>)</span>
<b>by </b><i>, <a class="txt" href="goboard5.html#E14:41"><i><font color="Green">E54</font></i></a>, <a class="ref" href="goboard5.html#T3" target="_self">Th3</a>, <a class="txt" href="goboard5.html#E15:41"><i><font color="Green">E57</font></i></a>, <a class="ref" href="goboard5.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="goboard5.html#D2" target="_self">Def2</a>, <a class="txt" href="goboard5.html#E1:41"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard5.html#T4" target="_self">Th4</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_4"><b>suppose </b></a><a NAME="E1:41_2_4"/><i><font color="Green">E83</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"><a NAME="E2:41_2_4"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">j2</font> <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="txt" href="goboard5.html#E10:41"><i><font color="Green">E46</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:41_2_4"/>
(  <a href="goboard5.html#K5" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3" target="_self">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">i1</font>,<font color="Maroon">j2</font> &amp;  <a href="goboard5.html#K4" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3" target="_self">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">j2</font> )
 <b>by </b><i><a class="ref" href="goboard5.html#D2" target="_self">Def2</a>, <a class="txt" href="goboard5.html#E1:41"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="goboard5.html#T4" target="_self">Th4</a>, </i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="goboard5.html#K5" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> = 
 <a href="topreal1.html#K3">LSeg</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">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<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">i1</font>,<font color="Maroon">j2</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E11:41"><i><font color="Green">E47</font></i></a>, , , <a class="txt" href="goboard5.html#E15:41"><i><font color="Green">E57</font></i></a>, </i>
<br/>.= 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="goboard5.html#D2" target="_self">Def2</a>, <a class="txt" href="goboard5.html#E1:41"><i><font color="Green">E41</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green">E43</font></i></a>, <a class="ref" href="goboard5.html#T4" target="_self">Th4</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
