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<b>let </b><font color="Maroon">z1</font> be  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">z2</font> be  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>set </b><font color="Maroon">r1</font> =  <a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z1</font>;<br/>
<b>set </b><font color="Maroon">r2</font> =  <a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z2</font>;<br/>
<b>set </b><font color="Maroon">i1</font> =  <a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z1</font>;<br/>
<b>set </b><font color="Maroon">i2</font> =  <a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z2</font>;<br/>
<a NAME="E1:93"/><i><font color="Green">E21</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z1</font></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" target="_self">Im</a> <font color="Maroon">z1</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z2</font></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" target="_self">Im</a> <font color="Maroon">z2</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<i><font color="Green">E22</font></i>: <span class="p1">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="txt" href="complex1.html#E11"><i><font color="Green">Lemma32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p5">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<span class="p1">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z2</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i><a class="txt" href="complex1.html#E11"><i><font color="Green">Lemma32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z2</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z1</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>

;<br/>
<a NAME="E4:93"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z2</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" target="_self">Im</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z1</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z1</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z2</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z2</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="complex1.html#E39"><i><font color="Green">Lemma69</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:93"/>
<span class="p1"><a href="complex1.html#K17" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z2</font></span>)</span></span><a href="complex1.html#K17" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17" target="_self">|.</a><span class="default"><font color="Maroon">z1</font></span><a href="complex1.html#K17" target="_self">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17" target="_self">|.</a><span class="default"><font color="Maroon">z2</font></span><a href="complex1.html#K17" target="_self">.|</a></span>
 <b>by </b><i><a class="txt" href="complex1.html#E38"><i><font color="Green">Lemma68</font></i></a>, <a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>;<br/>


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