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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">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">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:40"/><i><font color="Green">E52</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> )
 <b>and </b><br/><a NAME="E2:40"/><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>
 <b>and </b><br/><a NAME="E3:40"/><i><font color="Green">E59</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">n</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_1_1"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K4" target="_self">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font> &amp;  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K3" target="_self">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><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="E3:40_1_2"/><i><font color="Green">E61</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> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</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="E4:40_1_2"/><i><font color="Green">E62</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> &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> <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="E5:40_1_2"/><i><font color="Green">E63</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="E6:40_1_2"/><i><font color="Green">E64</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">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">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">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/><b>set </b><font color="Maroon">lf</font> =  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font>;<br/><b>set </b><font color="Maroon">lfn</font> =  <a href="gobrd13.html#K4" target="_self">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font>;<br/><b>set </b><font color="Maroon">rf</font> =  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font>;<br/><b>set </b><font color="Maroon">rfn</font> =  <a href="gobrd13.html#K3" target="_self">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font>;<br/><a NAME="E7:40_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a></i>;<br/><a NAME="E8:40_1_2"/><i><font color="Green">E66</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E9:40_1_2"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> &amp; <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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T3">GOBOARD2:3</a></i>;<br/><a NAME="E10:40_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <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> <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> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_2_1_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="40_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_2_1_1_1"/><i><font color="Green">E68</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"><b>hence </b> <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> = 
 <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>
<b>by </b><i>, , , , , </i>
<br/>.= 
 <a href="gobrd13.html#K4" target="_self">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
<b>by </b><i>, , , , , , , , , </i>
;<br/><br/><b>thus </b> <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> = 
 <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font>
<b>by </b><i>, , , , , , </i>
<br/>.= 
 <a href="gobrd13.html#K3" target="_self">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
<b>by </b><i>, , , , , , , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_2_1_1_2"/><i><font color="Green">E69</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"><b>hence </b> <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> = 
 <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font>
<b>by </b><i>, , , , , </i>
<br/>.= 
 <a href="gobrd13.html#K4" target="_self">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
<b>by </b><i>, , , , , , , , , </i>
;<br/><br/><b>thus </b> <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> = 
 <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>
<b>by </b><i>, , , , , , </i>
<br/>.= 
 <a href="gobrd13.html#K3" target="_self">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
<b>by </b><i>, , , , , , , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:40_1_2_1_1_3"/><i><font color="Green">E78</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"><b>hence </b> <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> = 
 <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>
<b>by </b><i>, , , , , </i>
<br/>.= 
 <a href="gobrd13.html#K4" target="_self">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
<b>by </b><i>, , , , , , , , , </i>
;<br/><br/><b>thus </b> <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> = 
 <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font>
<b>by </b><i>, , , , , , </i>
<br/>.= 
 <a href="gobrd13.html#K3" target="_self">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
<b>by </b><i>, , , , , , , , , </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:40_1_2_1_1_4"/><i><font color="Green">E79</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"><b>hence </b> <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> = 
 <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i1</font>,<font color="Maroon">j2</font>
<b>by </b><i>, , , , , </i>
<br/>.= 
 <a href="gobrd13.html#K4" target="_self">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
<b>by </b><i>, , , , , , , , , </i>
;<br/><br/><b>thus </b> <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> = 
 <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>
<b>by </b><i>, , , , , , </i>
<br/>.= 
 <a href="gobrd13.html#K3" target="_self">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font>
<b>by </b><i>, , , , , , , , , </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:40_1_2"/>
(  <a href="gobrd13.html#K4" target="_self">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K4" target="_self">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font> &amp;  <a href="gobrd13.html#K3" target="_self">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K3" target="_self">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">G</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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