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<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E23</font></i>: 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i/>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="complex2.html#T8" target="_self">Th8</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 

;<br/><b>hence </b><a NAME="E3:48_1"/>
(  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complex1.html#K17">.|</a></span> or  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complex1.html#K17">.|</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="square_1.html#T109">SQUARE_1:109</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:48_2"/>
(  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complex1.html#K17">.|</a></span> or  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complex1.html#K17">.|</a></span></span>)</span> )
 ;<br/><b>then </b><span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">x</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">y</font></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="square_1.html#K5">^2</a> 

<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex2.html#T8" target="_self">Th8</a></i>
;<br/><b>hence </b><a NAME="E3:48_2"/>
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complex2.html#K1" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/></div>
<b>end;</b></div>


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