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<a NAME="E1:28_1_1"/><i><font color="Green" title="E20">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</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">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></i>;<br/>
<i><font color="Green" title="E21">A2</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp1.html#K4" title="COMPLSP1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></span>)</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">z</font> <a href="complsp1.html#K4" title="COMPLSP1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T3" target="_self" title="COMPLSP2:th.3">Th3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</font>
<b>by </b><i><a class="txt" href="#E1:28_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="complsp2.html#T7" target="_self" title="COMPLSP2:th.7">Th7</a></i>
;<br/>
<a NAME="E3:28_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp1.html#K4" title="COMPLSP1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp1.html#K4" title="COMPLSP1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/>

<b>assume </b><a NAME="E1:28_1_1_2"/>
<font color="Maroon" title="c2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp1.html#K4" title="COMPLSP1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:28_1_1_2"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp1.html#K4" title="COMPLSP1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp1.html#K4" title="COMPLSP1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c3">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:28_1_1_2"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c3">x</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"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp1.html#K4" title="COMPLSP1:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:28_1_1_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">i</font> = <font color="Maroon" title="c3">x</font> as    <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>  <b>by </b><i><a class="txt" href="#E3:28_1_1_2"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/>
<a NAME="E6:28_1_1_2"/><i><font color="Green" title="E24">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">z</font> )
 
<b>by </b><i><a class="txt" href="#E2:28_1_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="#E4:28_1_1_2"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp1.html#K4" title="COMPLSP1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp1.html#K4" title="COMPLSP1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T16" target="_self" title="COMPLSP2:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:28_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="complsp2.html#T25" target="_self" title="COMPLSP2:th.25">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_2.html#K4" title="BINOP_2:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E6:28_1_1_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="complsp2.html#D1" target="_self" title="COMPLSP2:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T27" target="_self" title="COMPLSP2:th.27">Th27</a></i>
;<br/>
<b>hence </b><a NAME="E8:28_1_1_2"/>
<font color="Maroon" title="c2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="#E3:28_1_1_2"><i><font color="Green" title="E22">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:28_1_1"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp1.html#K4" title="COMPLSP1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
