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<a NAME="E1:32_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E33</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complsp1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E3:32_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complsp1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_1_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complsp1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_1_1_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complsp1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complsp1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:32_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complsp1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:32_1_1_2"/><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="complsp2.html#D1" target="_self">Def1</a>, </i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complsp1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complsp1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E8:32_1_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_1_1"/>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="complsp1.html#K6">*</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complsp2.html#K1" target="_self">*'</a> </span>)</span></span>)</span> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>


</div>
