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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> , <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>assume </b><a NAME="E1:39"/><i><font color="Green">E21</font></i>: 
(  <a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>thus </b> <a href="complex1.html#K3" target="_self">Re</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z2</font></span>)</span> = 
<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="real_1.html#K4">*</a> <span class="p2">(<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="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="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z2</font></span>)</span></span>)</span>
<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"><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="p1">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z2</font></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b> <a href="complex1.html#K4" target="_self">Im</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">z2</font></span>)</span> = 
<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="real_1.html#K4">*</a> <span class="p2">(<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="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="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complex1.html#E11"><i><font color="Green">Lemma32</font></i></a></i>
<br/>.= 
0
<b>by </b><i/>
;<br/>


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