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<b>let </b><font color="Maroon" title="c1">z</font> be    <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> ;<br/>

<a NAME="E1:64"/><i><font color="Green" title="E53">A1</font></i>: 
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T112" target="_self" title="COMPLEX1:th.112">Th112</a></i>;<br/>
<b>thus </b> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> = 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T112" target="_self" title="COMPLEX1:th.112">Th112</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex1.html#E1:64"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="complex1.html#T64" target="_self" title="COMPLEX1:th.64">Th64</a></i>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T64" target="_self" title="COMPLEX1:th.64">Th64</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="complex1.html#D5" target="_self" title="COMPLEX1:def.5">COMPLEX1:def 5</a><br/>

<b>thus </b> <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> = 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T112" target="_self" title="COMPLEX1:th.112">Th112</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex1.html#E1:64"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="complex1.html#T64" target="_self" title="COMPLEX1:th.64">Th64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><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="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T188" title="XCMPLX_1:th.188">XCMPLX_1:188</a></i>
<br/>.= 
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T64" target="_self" title="COMPLEX1:th.64">Th64</a></i>
;<br/>


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