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<b>set </b><font color="Maroon">B</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/>
<a NAME="E1:56_1_1"/><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E2:56_1_1"/><i><font color="Green">E54</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E3:56_1_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:56_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 ;<br/>

<a NAME="E2:56_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:56_1_1_1"/>
 ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> )
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">C</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> as   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="matrix_1.html#T9">MATRIX_1:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<a NAME="E5:56_1_1"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">C</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:56_1_1"/><b>then </b>
<font color="Maroon">C</font> is    <a href="matrix_1.html#M1">Matrix</a> of 1, <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>, <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:56_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a>, </i>;<br/>

<b>thus </b><a NAME="E8:56_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>

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

<b>assume </b><a NAME="E9:56_1_1"/><i><font color="Green">E56</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">x</font>
 ;<br/>

<a NAME="E10:56_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:56_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:56_1_1"/>
<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="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:56_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<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> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">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/>
<a NAME="E15:56_1_1"/>
<font color="Maroon">x</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<b>hence </b><a NAME="E16:56_1_1"/>
<font color="Maroon">C</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a>, , <a class="ref" href="matrixr1.html#T1" target="_self">Th1</a></i>;<br/>


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