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<b>let </b><font color="Maroon" title="c1">z1</font>, <font color="Maroon" title="c2">z2</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/>



<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E48">A1</font></i>: 
(  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c2">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green" title="E49">A2</font></i>: 
( <font color="Maroon" title="c1">z1</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c2">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">z1</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> &amp;  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="complex1.html#T80" target="_self" title="COMPLEX1:th.80">Th80</a>, <a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/>
<b>hence </b> <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c2">z2</font></span>)</span> = 
 <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> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="c2">z2</font> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex1.html#E1:53"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="complex1.html#T31" target="_self" title="COMPLEX1:th.31">Th31</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> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex1.html#E1:53"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="complex1.html#T80" target="_self" title="COMPLEX1:th.80">Th80</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c2">z2</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
;<br/><br/>

<b>thus </b><a NAME="E4:53"/>
 <a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="complex1.html#E1:53"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="complex1.html#E2:53"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="complex1.html#T31" target="_self" title="COMPLEX1:th.31">Th31</a></i>;<br/>


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