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<b>let </b><font color="Maroon">z</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">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<i><font color="Green">E53</font></i>: <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">p</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font> = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">p</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">p</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/>
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">p</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font> = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">p</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">p</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/>
<b>hence </b><a NAME="E3:102"/>
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">p</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default">0,<font color="Maroon">p</font></span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> &amp;  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">p</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">p</font>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>


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