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

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">M2</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E29</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> )
 ;<br/>

<a NAME="E2:25"/>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>
<a NAME="E3:25"/><b>then </b>
<span class="p1">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:25"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E5:25"/><b>then </b><i><font color="Green">E39</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> implies (  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">q</font> <a href="matrix_5.html#K9" target="_self">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Olive">q</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">q</font> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> ) ) );<br/>
<b>set </b><font color="Maroon">q0</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>;<br/>
<a NAME="E6:25"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E7:25"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="matrix_5.html#K9" target="_self">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E8:25"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 holds <br/><span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="binop_2.html#K5">*</a> <span class="p4">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E9:25"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:25"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i2</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i2</font>]
 ;<br/>

<b>then consider </b><font color="Maroon">q2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> implies (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span> &amp; ( for <font color="Olive">k2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k2</font> &amp; <font color="Olive">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Olive">k2</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Olive">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> ) ) )
 ;<br/>
<b>set </b><font color="Maroon">k0</font> = <font color="Maroon">i2</font>;<br/>
<b>set </b><font color="Maroon">q3</font> = <font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p4">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>;<br/>
<a NAME="E4:25_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> implies (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p0">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p0">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green">E45</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1
 </span><b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1_1"/><i><font color="Green">E56</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><i><font color="Green">E57</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, , <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><a NAME="E3:25_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, </i>;<br/><a NAME="E4:25_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E5:25_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E6:25_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:25_1_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:25_1_1_1_1"/>
for <font color="Olive">k2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k2</font> &amp; <font color="Olive">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p0">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p0">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Olive">k2</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Olive">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_1_1_1_2"/><i><font color="Green">E71</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 )
 ;<br/>

<a NAME="E2:25_1_1_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1_1_1_2_1"/>
( <font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1 or <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1_1_2_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:25_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, </i>;<br/><a NAME="E4:25_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E5:25_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:25_1_1_1_1_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:25_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, </i>;<br/><a NAME="E8:25_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E9:25_1_1_1_1_2_1_1"/>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:25_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E11:25_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E12:25_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:25_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, </i>;<br/><a NAME="E14:25_1_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E15:25_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p0">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p0">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, , , , , <a class="ref" href="matrix_5.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_1_1_2_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:25_1_1_1_1_2_1_2"/>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, </i>;<br/><a NAME="E3:25_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, </i>;<br/><a NAME="E4:25_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">k2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E5:25_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:25_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, </i>;<br/><a NAME="E7:25_1_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E8:25_1_1_1_1_2_1_2"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p4">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> <a href="matrix_5.html#K9" target="_self">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E9:25_1_1_1_1_2_1_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E10:25_1_1_1_1_2_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p4">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><b>hence </b><a NAME="E11:25_1_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p0">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p0">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Maroon">k2</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">k2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, , , <a class="ref" href="matrix_5.html#D7" target="_self">Def7</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E9:25_1_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p5">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p5">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q2</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p0">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p0">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, , , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:25_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i2</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:25"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E12:25"/>
 ex <font color="Olive">s</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> &amp; <font color="Olive">s</font> <a href="matrix_5.html#K9" target="_self">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j</font></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> holds <br/><font color="Olive">s</font> <a href="matrix_5.html#K9" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">s</font> <a href="matrix_5.html#K9" target="_self">.</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>


</div>
