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<span class="kw">let </span><font color="Maroon" title="c1">F</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a>  )</span><br/>

<a NAME="E1:30"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> <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">F</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#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:30"/><span class="lab"><font color="Green" title="E23">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="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">f22</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E4:30"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c2">f22</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</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> 1
 <span class="kw">and </span><br/><a NAME="E5:30"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> holds <br/><font color="Maroon" title="c2">f22</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f22</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</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> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">and </span><br/><a NAME="E6:30"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f22</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T1" title="FINSOP_1:th.1">FINSOP_1:1</a></span>;<br/>
<a NAME="E7:30"/><span class="lab"><font color="Green" title="E27">A6</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="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f22</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> $1;<br/>
<a NAME="E8:30"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being    <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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
;<br/>
<a NAME="E9:30"/>
 ex <font color="Olive" title="b1">f2</font> being    <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>  st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>,<font color="Olive" title="b1">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b2">k</font>] ) )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E28">A7</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">f2</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E11:30"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:30"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">k</font>]
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">f9</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a><span class="kw"> such that </span><br/><a NAME="E14:30"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</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="c3">f2</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> holds <br/><font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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="Olive" title="b1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#T21" target="_self" title="MATRIXC1:th.21">Th22</a></span>;<br/>
<a NAME="E15:30"/><span class="lab"><font color="Green" title="E32">A11</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="c3">f2</font> <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> <font color="Maroon" title="c3">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></span>;<br/>
<a NAME="E16:30"/><span class="kw">then </span>
1 <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="c3">f2</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="#E2:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E17:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30"><span class="lab"><font color="Green" title="E31">A10</font></span></a></span>;<br/>
<a NAME="E18:30"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f2</font> <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">F</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="#E11:30"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E19:30"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> holds <br/><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</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> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</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"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> implies <font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c1">F</font> <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"> <font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)</span><br/>

<a NAME="E3:30_1"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c1">F</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="#E2:30_1"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:30_1"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A19</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E6:30_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E7:30_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<font color="Maroon" title="c2">f22</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E30">A9</font></span></a></span>
<br/>.= 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f22</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="txt" href="#E5:30_1"><span class="lab"><font color="Green" title="E40">A19</font></span></a></span>
<br/>.= 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E5:30_1"><span class="lab"><font color="Green" title="E40">A19</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:30_1"/>
<font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</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><span class="kw">end;</span></div>
<a NAME="E20:30"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</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="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</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="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_2"/><span class="lab"><font color="Green" title="E37">A21</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">and </span><br/><a NAME="E2:30_2"/><span class="lab"><font color="Green" title="E38">A22</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font>
 ; <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="c3">f2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</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="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)</span><br/>

<a NAME="E3:30_2"/><span class="lab"><font color="Green" title="E39">A23</font></span>: 
<font color="Maroon" title="c5">n</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> <font color="Maroon" title="c3">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E2:30_2"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:30_2"/><span class="lab"><font color="Green" title="E40">A24</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:30_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E6:30_2"/><span class="lab"><font color="Green" title="E42">A26</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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> <font color="Maroon" title="c1">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_2"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E7:30_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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> <font color="Maroon" title="c3">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_2"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</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> <span class="p3">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)</span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E19:30"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="txt" href="#E2:30_2"><span class="lab"><font color="Green" title="E38">A22</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span>)</span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_2"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E6:30_2"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></span>
<br/>.= 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#T20" target="_self" title="MATRIXC1:th.20">Th21</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:30_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f2</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="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_2"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="txt" href="#E3:30_2"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</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><span class="kw">end;</span></div>
<a NAME="E21:30"/><span class="lab"><font color="Green" title="E37">A27</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> holds <br/><font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> implies <font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_3"/><span class="lab"><font color="Green" title="E38">A28</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">and </span><br/><a NAME="E2:30_3"/><span class="lab"><font color="Green" title="E39">A29</font></span>: 
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)</span><br/>

<a NAME="E3:30_3"/><span class="lab"><font color="Green" title="E40">A30</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:30_3"/>
<font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c3">f2</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="#E1:30"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E18:30"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E15:30"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E2:30_3"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:30_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A31</font></span>: 
<font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E6:30_3"/>
<font color="Maroon" title="c5">n</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="c3">f2</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="#E1:30"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E18:30"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E2:30_3"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></span>;<br/>
<a NAME="E7:30_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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="c5">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E3:30_3"><span class="lab"><font color="Green" title="E40">A30</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:30_3"/>
<font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:30"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="txt" href="#E1:30_3"><span class="lab"><font color="Green" title="E38">A28</font></span></a>, <a class="txt" href="#E2:30_3"><span class="lab"><font color="Green" title="E39">A29</font></span></a>, <a class="txt" href="#E5:30_3"><span class="lab"><font color="Green" title="E41">A31</font></span></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><span class="kw">end;</span></div>
<a NAME="E22:30"/><span class="lab"><font color="Green" title="E38">A32</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">d</font> = <span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> ;<br/>
<a NAME="E23:30"/><span class="lab"><font color="Green" title="E39">A33</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E18:30"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E27">A6</font></span></a></span>
;<br/>
<a NAME="E25:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A34</font></span>: 
<span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f9</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E18:30"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E14:30"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E15:30"><span class="lab"><font color="Green" title="E32">A11</font></span></a></span>;<br/>
<font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1</span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">F</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> 1</span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:30"><span class="lab"><font color="Green" title="E39">A33</font></span></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> 1</span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E22:30"><span class="lab"><font color="Green" title="E38">A32</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f2</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> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E18:30"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></span>
;<br/>
<a NAME="E27:30"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c1">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:30"><span class="lab"><font color="Green" title="E39">A33</font></span></a>, <a class="txt" href="#E17:30"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E21:30"><span class="lab"><font color="Green" title="E37">A27</font></span></a>, <a class="txt" href="#E25:30"><span class="lab"><font color="Green" title="E40">A34</font></span></a>, <a class="ref" href="finsop_1.html#T2" title="FINSOP_1:th.2">FINSOP_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:30"/>
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
 ; <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>
