<?xml version="1.0"?>
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<b>let </b><font color="Maroon">X</font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>





<a NAME="E1:70"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E2:70"/>
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="csspace.html#K12" target="_self">.|.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:70"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="csspace.html#K12" target="_self">.|.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="csspace.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:70"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E5:70"/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="csspace.html#K12" target="_self">.|.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<a NAME="E6:70"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="csspace.html#K12" target="_self">.|.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i>, <a class="ref" href="csspace.html#T3" target="_self">Th3</a>, <a class="ref" href="square_1.html#T96">SQUARE_1:96</a></i>;<br/>
<i><font color="Green">E38</font></i>:  <a href="complex1.html#K3">Re</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"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T19">COMPLEX1:19</a></i>
<br/>.= 
<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"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T19">COMPLEX1:19</a></i>
<br/>.= 
<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> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T19">COMPLEX1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="csspace.html#D4" target="_self">Def4</a></i>
;<br/>
<i><font color="Green">E81</font></i>:  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> = 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T112">COMPLEX1:112</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/>
<i><font color="Green">E93</font></i>:  <a href="complex1.html#K4">Im</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"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T19">COMPLEX1:19</a></i>
<br/>.= 
<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"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T19">COMPLEX1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T19">COMPLEX1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<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">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i/>
;<br/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> = 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T112">COMPLEX1:112</a></i>
<br/>.= 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E11:70"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="complex1.html#K4">Im</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"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="csspace.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E12:70"/>
 <a href="complex1.html#K3">Re</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"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="csspace.html#K12" target="_self">.|.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E13:70"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="complex1.html#K3">Re</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"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E14:70"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="csspace.html#E12"><i><font color="Green">Lemma67</font></i></a>, <a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/>
<a NAME="E15:70"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="complex1.html#K8">+</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p4">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="complex1.html#K8">+</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i>, , , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E16:70"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="csspace.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E17:70"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E18:70"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E19:70"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> 
 
;<br/>
<a NAME="E20:70"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E21:70"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E22:70"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> 
 
;<br/>
<a NAME="E23:70"/>
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> )
 
<b>by </b><i>, <a class="ref" href="complex1.html#T140">COMPLEX1:140</a></i>;<br/>
<a NAME="E24:70"/><b>then </b>
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E25:70"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E26:70"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p3"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></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"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="csspace.html#T6" target="_self">Th6</a>, <a class="ref" href="csspace.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E27:70"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></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">2 <a href="real_1.html#K4">*</a> <span class="p4"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E28:70"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p2"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span></span>)</span> <a href="square_1.html#K5">^2</a> 
 
;<br/>
<a NAME="E29:70"/>
( <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E30:70"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E31:70"/>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">y</font></span><a href="csspace.html#K13" target="_self">.||</a></span>
 <b>by </b><i><a class="ref" href="csspace.html#T8" target="_self">Th8</a>, <a class="ref" href="square_1.html#T78">SQUARE_1:78</a></i>;<br/>


</div>
