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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  ;<br/>
<i><font color="Green">E10</font></i>: <span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D3">COMSEQ_3:def 3</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D5">COMSEQ_1:def 5</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>3</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>3</sub></font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T126">COMPLEX1:126</a></i>
;<br/>
<a NAME="E3:9"/>
( 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>3</sub></font></span>)</span> <a href="square_1.html#K5">^2</a>  &amp; 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>3</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="E4:9"/><b>then </b>
<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <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="txt" href="csspace2.html#E2:9"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E5:9"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K3">Re</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>

<span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="comseq_3.html#D4">COMSEQ_3:def 4</a></i>
<br/>.= 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_1.html#D5">COMSEQ_1:def 5</a></i>
<br/>.= 
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="comseq_2.html#D2">COMSEQ_2:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E7:9"/>
<span class="p1">(<span class="default"><a href="comseq_3.html#K4">Im</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_1.html#K3">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="comseq_2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T126">COMPLEX1:126</a></i>;<br/>


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