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<div class="add">

<b>thus </b><a NAME="E1:1_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>3</sub></font> ) &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>3</sub></font> ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_2_1"/><i><font color="Green">E1</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K15">*'</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="complex1.html#K15">*'</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<a NAME="E2:1_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_2_1_1"/><i><font color="Green">E2</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K15">*'</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="complsp2.html#E1:1_1_2_1"><i><font color="Green">E1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:1_1_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="complsp2.html#E1:1_1_2_1"><i><font color="Green">E1</font></i></a>, <a class="txt" href="complsp2.html#E1:1_1_2_1_1"><i><font color="Green">E2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:1_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="complsp2.html#E1:1_1_2_1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>


</div>
