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

<b>let </b><font color="Maroon">f</font> be  non <a href="xboole_0.html#V1">empty</a> <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>set </b><font color="Maroon">x</font> =  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">y</font> =  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font>;<br/>
<a NAME="E1:35"/><i><font color="Green">E50</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" target="_self">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" target="_self">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="goboard2.html#K3" target="_self">GoB</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="goboard2.html#K1" target="_self">GoB</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E2:35"/><i><font color="Green">E51</font></i>: 
(  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" target="_self">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="goboard2.html#K3" target="_self">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K3" target="_self">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="goboard2.html#K2" target="_self">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="goboard2.html#K2" target="_self">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<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>assume </b><a NAME="E3:35"/><i><font color="Green">E56</font></i>: 
<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>
 ;<br/>

<a NAME="E4:35"/><i><font color="Green">E57</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</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">f</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</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">X_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</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#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</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> ;<br/>
<a NAME="E6:35"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E7:35"/><b>then </b><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</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="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> <a href="goboard1.html#K1">.</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="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E8:35"/><i><font color="Green">E60</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>then 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="E10:35"/><i><font color="Green">E63</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="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <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="finseq_2.html#T11">FINSEQ_2:11</a></i>;<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="E12:35"/><i><font color="Green">E64</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="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">i</font>
;<br/>

<b>take </b>
<font color="Maroon">j</font>
;<br/>

<a NAME="E13:35"/>
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" target="_self">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <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="goboard2.html#K3" target="_self">GoB</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" target="_self">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f</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#T31">FINSEQ_3:31</a></i>;<br/>
<b>hence </b><a NAME="E14:35"/>
<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" target="_self">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>

<a NAME="E15:35"/><b>then </b>
( <span class="p1">(<span class="default"><a href="goboard2.html#K3" target="_self">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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 
<b>by </b><i>, , , <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<b>hence </b><a NAME="E16:35"/>
<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" target="_self">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>
 ;<br/>


</div>
