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

<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">i</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">m</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>set </b><font color="Maroon">D</font> =  <a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>;<br/>
<b>assume </b><a NAME="E1:41"/><i><font color="Green">E39</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 ;<br/>

<b>then consider </b><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="E3:41"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">m</font></span>)</span> &amp; <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="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:41"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:41"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green">E42</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">m</font>,<font color="Maroon">j</font> &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> <font color="Maroon">G</font> &amp; 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="matrix_1.html#K1">width</a> <font color="Maroon">G</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E7:41"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">M</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="E9:41"/><i><font color="Green">E44</font></i>: 
(  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i>, </i>;<br/>
<a NAME="E10:41"/><i><font color="Green">E47</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_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"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</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="41_1_1"><b>suppose </b></a><a NAME="E1:41_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:41_1_1"/><b>then </b><i><font color="Green">E49</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="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">j</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E3:41_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:41_1_1"/><b>then </b>
<font color="Maroon">j</font> <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/><a NAME="E5:41_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:41_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><b>hence </b><a NAME="E7:41_1_1"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_2"><b>suppose </b></a><a NAME="E1:41_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:41_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">i</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="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1 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="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E4:41_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">l</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="xreal_1.html#T21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E5:41_1_2"/><b>then </b><i><font color="Green">E52</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="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">l</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> )
 <b>by </b><i>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:41_1_2"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="goboard1.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:41_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="goboard1.html#E1:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><b>hence </b><a NAME="E8:41_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4" target="_self">DelCol</a> <font color="Maroon">G</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">m</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
