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

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">aa</font> = <font color="Maroon" title="c2">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></span>;<br/>
<a NAME="E2:16"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T3" target="_self" title="COMPLSP2:th.3">Th3</a></span>;<br/>
<a NAME="E3:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></span>;<br/>
<a NAME="E4:16"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T3" target="_self" title="COMPLSP2:th.3">Th3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E14">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E2:16_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span><br/><a NAME="E3:16_1"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">aa</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T16" target="_self" title="COMPLSP2:th.16">Th16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">aa</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T35" title="COMPLEX1:th.35">COMPLEX1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T16" target="_self" title="COMPLSP2:th.16">Th16</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:16"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="complsp2.html#T3" target="_self" title="COMPLSP2:th.3">Th3</a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
