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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><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/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:62"/><i><font color="Green">E51</font></i>: 
( 0 <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> <a href="square_1.html#K5">^2</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a>, <a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E2:62"/><i><font color="Green">E52</font></i>: 
0 <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="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<b>thus </b><span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp1.html#K14" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp1.html#K14" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><span class="p4"><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="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp1.html#T62" target="_self">Th62</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><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="square_1.html#K5">^2</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T84">RVSUM_1:84</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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="square_1.html#K5">^2</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3"><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="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complsp1.html#E1:62"><i><font color="Green">E51</font></i></a>, <a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>
<br/>.= 
<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="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="complsp1.html#E2:62"><i><font color="Green">E52</font></i></a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>
<br/>.= 
<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="binop_2.html#K11">*</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complsp1.html#K16" target="_self">.|</a></span>

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