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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp;  <a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c1">x</font> )</span><br/>

<a NAME="E1:56"/><span class="lab"><font color="Green" title="E42">A1</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">x</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="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></span>;<br/>
<span class="lab"><font color="Green" title="E43">A2</font></span>:  <a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T17" target="_self" title="COMPLSP2:th.17">Th17</a>, <a class="ref" href="complex1.html#T31" title="COMPLEX1:th.31">COMPLEX1:31</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p2">(<span class="default"><a href="seq_4.html#K11" title="SEQ_4:func.11">-</a> <span class="p3">(<span class="default"><a href="seq_4.html#K11" title="SEQ_4:func.11">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T54" target="_self" title="COMPLSP2:th.54">Th54</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="complsp2.html#T30" target="_self" title="COMPLSP2:th.30">Th30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T53" target="_self" title="COMPLSP2:th.53">Th53</a></span>
<br/>.= 
 <a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c1">x</font>

;<br/>
 <a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T17" target="_self" title="COMPLSP2:th.17">Th17</a>, <a class="ref" href="complex1.html#T31" title="COMPLEX1:th.31">COMPLEX1:31</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T54" target="_self" title="COMPLSP2:th.54">Th54</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="complsp2.html#T43" target="_self" title="COMPLSP2:th.43">Th43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T53" target="_self" title="COMPLSP2:th.53">Th53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K13" title="BINOP_2:func.13">-</a> <span class="p4">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span></span>)</span> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K10" title="SEQ_4:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a></span>)</span></span>)</span>

<br/>.= 
 <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complsp2.html#T53" target="_self" title="COMPLSP2:th.53">Th53</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:56"/>
(  <a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<span class="default"><a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp;  <a href="complsp2.html#K7" title="COMPLSP2:func.7">Im</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="seq_4.html#K12" title="SEQ_4:func.12">*</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complsp2.html#K6" title="COMPLSP2:func.6">Re</a> <font color="Maroon" title="c1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E43">A2</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/>


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