<?xml version="1.0"?>
<div class="add">

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





<b>set </b><font color="Maroon">ab</font> = <font color="Maroon">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">rab</font> =  <a href="complex2.html#K2" target="_self">Rotate</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">ra</font> =  <a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">a</font>,<font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">rb</font> =  <a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font>;<br/>
<a NAME="E1:54"/><i><font color="Green">E23</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <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">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">r</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">Im</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <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">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r</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#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex2.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E2:54"/><i><font color="Green">E26</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">a</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">r</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">Im</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">a</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r</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#K4">Im</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex2.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E3:54"/><i><font color="Green">E29</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">r</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">Im</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r</font></span>)</span></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r</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#K4">Im</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">r</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex2.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E4:54"/><i><font color="Green">E30</font></i>: 
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">b</font></span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T19">COMPLEX1:19</a></i>;<br/>
<i><font color="Green">E31</font></i>:  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">a</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">r</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">Im</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r</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"><a href="complex1.html#K3">Re</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">r</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">Im</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="complex1.html#T19">COMPLEX1:19</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">r</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">a</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r</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#K4">Im</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">r</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"><a href="complex1.html#K3">Re</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K21">sin</a> <font color="Maroon">r</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#K4">Im</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K24">cos</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="complex1.html#T19">COMPLEX1:19</a></i>
<br/>.= 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">r</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E7:54"/>
 <a href="complex2.html#K2" target="_self">Rotate</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K8">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">a</font>,<font color="Maroon">r</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><a href="complex2.html#K2" target="_self">Rotate</a> <font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="complex1.html#T9">COMPLEX1:9</a></i>;<br/>


</div>
