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

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

<b>assume </b><a NAME="E1:31"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E3:31"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">d</font> = <span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> ;<br/>
<b>consider </b><font color="Maroon">f22</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E5:31"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f22</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> 1
 <b>and </b><br/><a NAME="E6:31"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</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">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> holds <br/><font color="Maroon">f22</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f22</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E7:31"/><i><font color="Green">E41</font></i>: 
<a href="binop_2.html#K27">addcomplex</a>  <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f22</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finsop_1.html#T2">FINSOP_1:2</a>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f22</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E8:31"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:31_1"/>
<font color="Maroon">f22</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> 
 
;<br/>
<b>hence </b><a NAME="E2:31_1"/>
( <font color="Maroon">k</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> implies  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">x</font>] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:31"/>
 ex <font color="Olive">f2</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#K4">dom</a> <font color="Olive">f2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">f2</font> <a href="matrix_5.html#K9">.</a> <font color="Olive">k</font>] ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i>
<b>then consider </b><font color="Maroon">f2</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="E11:31"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <font color="Olive">k</font>] ) )
 ;<br/>
<a NAME="E12:31"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/>
<a NAME="E13:31"/><i><font color="Green">E54</font></i>: 
1 <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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:31"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</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">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> holds <br/><font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:31_2"/><i><font color="Green">E56</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</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">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> )
 ;<br/>

<a NAME="E2:31_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:31_2"/><b>then </b><i><font color="Green">E61</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:31_2"/><i><font color="Green">E62</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="txt" href="matrixc1.html#E1:31"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E5:31_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:31_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">n</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:31"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E7:31_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E3:31"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrixc1.html#E2:31"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>then </b><font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<font color="Maroon">f22</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f22</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="matrixc1.html#E5:31"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="matrixc1.html#E5:31"><i><font color="Green">E38</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:31_2"/>
<font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:31"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green">E103</font></i>: <font color="Maroon">F</font> <a href="matrix_5.html#K9">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="matrix_5.html#K9">.</a> 1</span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K15">*'</a> 

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> 1</span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i>, <a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> 1</span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i>, , <a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> 1
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a>, , </i>
;<br/>
<a NAME="E17:31"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:31_4"/><i><font color="Green">E105</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:31_4"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:31"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E3:31_4"/><b>then </b><i><font color="Green">E106</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:31_4"/><b>then </b><i><font color="Green">E107</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</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="E5:31_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">n</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">F</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:31"><i><font color="Green">E37</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:31_4"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">n</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">f2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, , , <a class="txt" href="matrixc1.html#E6:31"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E7:31_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:31"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, , </i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="txt" href="matrixc1.html#E7:31"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:31"><i><font color="Green">E37</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="matrix_5.html#K9">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i>, <a class="txt" href="matrixc1.html#E6:31"><i><font color="Green">E40</font></i></a>, <a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>
<br/>.= 
<a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E9:31_4"/>
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E8:31"><i><font color="Green">E42</font></i></a>, <a class="txt" href="matrixc1.html#E5:31"><i><font color="Green">E38</font></i></a>, <a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E111</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i>, , </i>
;<br/>
<b>consider </b><font color="Maroon">f'</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E20:31"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> holds <br/><font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">n</font>
 <b>by </b><i/>;<br/>
<a NAME="E21:31"/><i><font color="Green">E114</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E22:31"/><i><font color="Green">E115</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, , , </i>;<br/>
<a NAME="E23:31"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> 1
 
<b>by </b><i><a class="txt" href="matrixc1.html#E5:31"><i><font color="Green">E38</font></i></a>, <a class="txt" href="matrixc1.html#E6:31"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E24:31"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:31_6"/><i><font color="Green">E118</font></i>: 
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 ;<br/>

<a NAME="E2:31_6"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E3:31_6"/><b>then </b><i><font color="Green">E119</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:31_6"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E11:31"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a>, </i>;<br/>
<a NAME="E5:31_6"/><i><font color="Green">E121</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E6:31_6"/>
<font color="Maroon">n</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> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E11:31"><i><font color="Green">E43</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, , , </i>;<br/>
<a NAME="E7:31_6"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</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">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E5:31"><i><font color="Green">E38</font></i></a>, <a class="ref" href="matrixc1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E8:31_6"/>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="ref" href="matrixc1.html#T2" target="_self">Th2</a>, <a class="txt" href="matrixc1.html#E5:31"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:31_6"/>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K27">addcomplex</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="matrix_5.html#K9">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E11:31"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrixc1.html#E2:31"><i><font color="Green">E36</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:31"/>
<span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="complex1.html#K15">*'</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:31"><i><font color="Green">E37</font></i></a>, , <a class="txt" href="matrixc1.html#E5:31"><i><font color="Green">E38</font></i></a>, , , </i>;<br/>
<a NAME="E26:31"/><b>then </b>
<span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="complex1.html#K15">*'</a>  <a href="hidden.html#R1">=</a> <a href="binop_2.html#K27">addcomplex</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="finsop_1.html#T3">FINSOP_1:3</a>, <a class="txt" href="matrixc1.html#E7:31"><i><font color="Green">E41</font></i></a>, <a class="txt" href="matrixc1.html#E1:31"><i><font color="Green">E35</font></i></a>, <a class="txt" href="matrixc1.html#E8:31"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E27:31"/>
<a href="binop_2.html#K27">addcomplex</a>  <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27">addcomplex</a>  <a href="finsop_1.html#K2">$$</a> <font color="Maroon">F</font></span>)</span> <a href="complex1.html#K15">*'</a> 
 ;<br/>


</div>
