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

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="nat_1.html#NM1">Nat</a>,  <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>means </b>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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> <font color="Maroon">v1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">v2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; ( for <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">v1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> &amp; <font color="Maroon">v2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> holds <br/><font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> ) );<br/>
<a NAME="E1:27_1_1"/><i><font color="Green">E51</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</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">v1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v2</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">v2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:27_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</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="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/>for <font color="Olive">p1</font>, <font color="Olive">p2</font> being  <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>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">p1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">p2</font>] holds <br/><font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Olive">p2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:27_1_1_1"/>
<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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</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="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font>, <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/>
<b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i'</font>, <font color="Maroon">s2</font> = <font color="Maroon">v2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j'</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>let </b><font color="Maroon">p1</font> be   <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/><b>let </b><font color="Maroon">p2</font> be   <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/>



<b>assume </b><a NAME="E4:27_1_1_1"/>
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">p1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">p2</font>] )
 ;<br/>

<a NAME="E5:27_1_1_1"/><b>then </b>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="euclid.html#K23">]|</a></span> )
 
;<br/>
<b>hence </b><a NAME="E6:27_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:27_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</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="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">p</font> being  <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> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:27_1_1_2"/><i><font color="Green">E58</font></i>: 
<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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</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="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font>, <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/>
<b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i'</font>, <font color="Maroon">s2</font> = <font color="Maroon">v2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j'</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>take </b>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="euclid.html#K23">]|</a></span>
;<br/>

<b>thus </b><a NAME="E4:27_1_1_2"/>
<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> <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> <font color="Maroon">v1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">v2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E5:27_1_1_2"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/>

<b>hence </b><a NAME="E6:27_1_1_2"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">M</font> being    <a href="matrix_1.html#M1">Matrix</a> of  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font>, <a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font>,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> such that </b><br/><a NAME="E5:27_1_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">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> <font color="Maroon">M</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="matrix_1.html#S2">MATRIX_1:sch 2</a>(, );<br/></i>
<a NAME="E6:27_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E7:27_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font> )
 
;<br/>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as   <a href="matrix_1.html#NM1">Matrix</a> of 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> ;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>thus </b><a NAME="E9:27_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font> )
 <b>by </b><i>, <a class="ref" href="matrix_1.html#T24">MATRIX_1:24</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>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>assume </b><a NAME="E10:27_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</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">M</font>
 ;<br/>

<b>hence </b><a NAME="E11:27_1_1"/>
<font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</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">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">v2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i/>;<br/>


</div>
