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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c</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">z</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>, <font color="Maroon">cc</font> = <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K14" target="_self">*</a> <font color="Maroon">z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i/>;<br/><a NAME="E3:57_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E4:57_1"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">ac</font> = <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="finseq_2.html#T13">FINSEQ_2:13</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K14" target="_self">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">cc</font></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i>, <a class="txt" href="complsp1.html#E23"><i><font color="Green">Lemma43</font></i></a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K5">*</a> <font color="Maroon">c'</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i>, <a class="ref" href="complsp1.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c'</font></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="binop_2.html#K11">*</a> <font color="Maroon">ac</font>
<b>by </b><i>, <a class="txt" href="complsp1.html#E23"><i><font color="Green">Lemma43</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="rvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T67">RVSUM_1:67</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:57"/>
 <a href="complsp1.html#K15" target="_self">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="complsp1.html#K14" target="_self">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c</font></span><a href="complex1.html#K17">.|</a></span> <a href="rvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>;<br/>


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