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<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<a NAME="E1:20"/><i><font color="Green">E11</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</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">c<sub>2</sub></font></span>)</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"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#D16">COMPLEX1:def 16</a></i>;<br/>
<a NAME="E2:20"/><i><font color="Green">E12</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E3:20"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E4:20"/><b>then </b>
0 <a href="real_1.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">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </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">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:20"/><b>then </b><i><font color="Green">E13</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  &amp; <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</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"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </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">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E6:20"/>
0 <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">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </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">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="comseq_3.html#E2:20"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:20"/><b>then </b><i><font color="Green">E14</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  &amp; <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</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"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </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">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E8:20"/><i><font color="Green">E15</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</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">c<sub>2</sub></font></span>)</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"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="comseq_3.html#E5:20"><i><font color="Green">E13</font></i></a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<a NAME="E9:20"/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</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">c<sub>2</sub></font></span>)</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"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="comseq_3.html#E7:20"><i><font color="Green">E14</font></i></a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<b>hence </b><a NAME="E10:20"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complex1.html#K17">.|</a></span> )
 <b>by </b><i><a class="txt" href="comseq_3.html#E1:20"><i><font color="Green">E11</font></i></a>, <a class="txt" href="comseq_3.html#E8:20"><i><font color="Green">E15</font></i></a>, <a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>;<br/>


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