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

<b>let </b><font color="Maroon">M</font> be  <a href="goboard1.html#V3" target="_self">X_equal-in-line</a> <a href="goboard1.html#V6" target="_self">X_increasing-in-column</a> <a href="matrix_1.html#NM4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:21"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">x</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">M</font>,<font color="Olive">n</font></span>)</span> &amp; <font color="Olive">x</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">M</font>,<font color="Olive">m</font></span>)</span> &amp; <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> &amp; <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> &amp; not <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <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="E3:21"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</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">M</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">x</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">M</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">M</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> )
 ;<br/>
<a NAME="E4:21"/><i><font color="Green">E40</font></i>: 
 <a href="goboard1.html#K2" target="_self">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">m</font></span>)</span> is <a href="funct_1.html#V3">constant</a>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Ln</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">n</font>, <font color="Maroon">Lm</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">m</font> as   <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>consider </b><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="E7:21"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ln</font> &amp; <font color="Maroon">Ln</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Mi</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font> as   <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>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="E10:21"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lm</font> &amp; <font color="Maroon">Lm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E11:21"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Ln</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="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Ln</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Lm</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="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Lm</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Ln</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Lm</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="goboard1.html#K2" target="_self">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>;<br/>
<a NAME="E12:21"/><i><font color="Green">E44</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Ln</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Ln</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Lm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">Lm</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E13:21"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> <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> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:21"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Ln</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ln</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Lm</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lm</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:21"/><b>then </b><i><font color="Green">E49</font></i>: 
(  <a href="goboard1.html#K2" target="_self">X_axis</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> is <a href="goboard1.html#V1" target="_self">increasing</a> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span></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="goboard1.html#K2" target="_self">X_axis</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span></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="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E16:21"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">Lm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">i</font> &amp; <font color="Maroon">Ln</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> &amp; <font color="Maroon">Lm</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">i</font> )
 
<b>by </b><i>, , , , , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E18:21"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Lm</font></span>)</span> &amp; <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="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Lm</font></span>)</span> )
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E19:21"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lm</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E20:21"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">Lm</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E21:21"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">Lm</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E22:21"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Lm</font></span>)</span> &amp; <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="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Lm</font></span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E23:21"/><b>then </b>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Lm</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Lm</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><i><font color="Green">E70</font></i>: <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  = 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <font color="Maroon">Lm</font></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , , , </i>
<br/>.= 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , </i>
;<br/>
<a NAME="E25:21"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E26:21"/>
<font color="Maroon">m</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#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E27:21"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Mi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E28:21"/>
<font color="Maroon">n</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#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E29:21"/><b>then </b>
<font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Mi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then </b><i><font color="Green">E74</font></i>: <span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, , , , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" target="_self">X_axis</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="goboard1.html#K1" target="_self">.</a> <font color="Maroon">m</font>
<b>by </b><i>, , , , , </i>
;<br/>
<a NAME="E31:21"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E32:21"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/>


</div>
