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<b>let </b><font color="Maroon">D</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">e</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>given </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:13_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> holds <br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:13_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">e</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Olive">i</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> <font color="Maroon">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:13_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font>
 ;<br/>

<a NAME="E2:13_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:13_1_1"/>
 ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">e</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</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> <font color="Maroon">n</font> )
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon">e</font> is   <a href="matrix_1.html#NM1">Matrix</a> of <font color="Maroon">D</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:13"/>
<font color="Maroon">e</font> is   <a href="matrix_1.html#NM1">Matrix</a> of <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:13"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">e</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Olive">i</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> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:13"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> holds <br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:13_2"/><i><font color="Green">E49</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">e</font>
 ;<br/>

<a NAME="E2:13_2"/>
 ex <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">e</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</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> <font color="Maroon">n</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:13_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
 ex <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">e</font> holds <br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">n</font>
 ;<br/>


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