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

<span class="kw">let </span><font color="Maroon" title="c1">cF</font> be   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">c</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  <font color="Olive" title="b1">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">cF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c1">cF</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">cF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c1">cF</font></span>)</span></span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">cF</font> being  <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a>  st  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">cF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Olive" title="b1">cF</font></span>)</span>;<br/>
<a NAME="E1:111"/><span class="lab"><font color="Green" title="E63">A2</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">k</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:111_1"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/>

<a NAME="E2:111_1"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</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/>
<span class="kw">let </span><font color="Maroon" title="c4">cF</font> be   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 implies <font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c4">cF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c4">cF</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:111_1"/><span class="lab"><font color="Green" title="E66">A5</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c4">cF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c4">cF</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">cF1</font> = <font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c4">cF</font>;<br/>
<a NAME="E4:111_1"/><span class="lab"><font color="Green" title="E67">A6</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c4">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c4">cF</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<a NAME="E5:111_1"/><span class="lab"><font color="Green" title="E68">A7</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">cF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c4">cF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">cF</font> = <font color="Maroon" title="c4">cF</font>, <font color="Maroon" title="c7">cF1</font> = <font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c4">cF</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="afinsq_2.html#E76"><span class="lab"><font color="Green" title="E50">Lm6</font></span></a></span>;<br/>
<a NAME="E7:111_1"/><span class="lab"><font color="Green" title="E69">A8</font></span>: 
<font color="Maroon" title="c6">cF</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</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> <font color="Maroon" title="c6">cF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></span>;<br/>
<a NAME="E8:111_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</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="#E3:111_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a></span>;<br/>
<a NAME="E9:111_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A9</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">cF</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111_1"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="afinsq_1.html#T14" title="AFINSQ_1:th.14">AFINSQ_1:14</a></span>;<br/>
<a NAME="E10:111_1"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</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="E11:111_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A10</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">cF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E12:111_1"/><span class="kw">then </span>
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">cF</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">cF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font></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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">cF</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</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="afinsq_2.html#T55" target="_self" title="AFINSQ_2:th.55">Th56</a></span>;<br/>
<a NAME="E13:111_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A11</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c6">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">cF</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">cF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111_1"><span class="lab"><font color="Green" title="E69">A8</font></span></a>, <a class="ref" href="binop_2.html#D3" title="BINOP_2:def.3">BINOP_2:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E73">A15</font></span>: <font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">cF</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> = 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">cF</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="txt" href="#E9:111_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a></span>
<br/>.= 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">cF1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T75" target="_self" title="AFINSQ_2:th.75">TTT1</a></span>
;<br/>
<a NAME="E15:111_1"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</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="E16:111_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">cF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E17:111_1"/><span class="lab"><font color="Green" title="E74">A16</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">cF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">cF1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" title="VALUED_1:th.6">VALUED_1:6</a></span>;<br/>
<a NAME="E18:111_1"/><span class="lab"><font color="Green" title="E75">A17</font></span>: 
<font color="Maroon" title="c7">cF1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</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> <font color="Maroon" title="c7">cF1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="txt" href="#E4:111_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></span>;<br/>
<a NAME="E19:111_1"/>
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">cF1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">cF1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font></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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">cF1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</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="#E4:111_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E11:111_1"><span class="lab"><font color="Green" title="E71">A10</font></span></a>, <a class="ref" href="afinsq_2.html#T55" target="_self" title="AFINSQ_2:th.55">Th56</a></span>;<br/>
<a NAME="E20:111_1"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">cF1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">cF1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">cF1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:111_1"><span class="lab"><font color="Green" title="E75">A17</font></span></a>, <a class="ref" href="binop_2.html#D3" title="BINOP_2:def.3">BINOP_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:111_1"/>
<font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c4">cF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c4">cF</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:111_1"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="txt" href="#E17:111_1"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="txt" href="#E14:111_1"><span class="lab"><font color="Green" title="E73">A15</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:111"/><span class="lab"><font color="Green" title="E64">A18</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">cF</font> be   <a href="valued_0.html#V1" title="VALUED_0:attr.1">complex-valued</a>  <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c3">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">cF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">cF</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:111_2"/><span class="lab"><font color="Green" title="E65">A19</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c3">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">cF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">cF</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">cF1</font> = <font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">cF</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">cF</font> = <font color="Maroon" title="c3">cF</font>, <font color="Maroon" title="c6">cF1</font> = <font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">cF</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="afinsq_2.html#E76"><span class="lab"><font color="Green" title="E50">Lm6</font></span></a></span>;<br/>
<a NAME="E3:111_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_2"><span class="lab"><font color="Green" title="E65">A19</font></span></a></span>;<br/>
<a NAME="E4:111_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A21</font></span>: 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c5">cF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#T1" title="BINOP_2:th.1">BINOP_2:1</a>, <a class="ref" href="afinsq_2.html#D9" target="_self" title="AFINSQ_2:def.9">Def9</a></span>;<br/>
<a NAME="E5:111_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">cF1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_2"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:111_2"/>
<font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c3">cF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">cF</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111_2"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="ref" href="binop_2.html#T1" title="BINOP_2:th.1">BINOP_2:1</a>, <a class="ref" href="afinsq_2.html#D9" target="_self" title="AFINSQ_2:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:111"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E64">A18</font></span></a>, <a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E63">A2</font></span></a>);<br/></span>
<a NAME="E4:111"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[ <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">cF</font>]
 
;<br/>
<span class="kw">hence </span><a NAME="E5:111"/>
<font color="Maroon" title="c2">c</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c1">cF</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c1">cF</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
