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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">z</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:56_1"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c2">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j</font>, <font color="Maroon" title="c5">c'</font> = <span class="p1">(<span class="default"><a href="complsp1.html#K12" title="COMPLSP1:func.12">-</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self" title="COMPLSP1:th.21">Th21</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="complsp1.html#K15" title="COMPLSP1:func.15">abs</a> <span class="p2">(<span class="default"><a href="complsp1.html#K12" title="COMPLSP1:func.12">-</a> <font color="Maroon" title="c2">z</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j</font> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c5">c'</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="txt" href="complsp1.html#E1:56_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="complsp1.html#T59" target="_self" title="COMPLSP1:th.59">Th59</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K1" title="BINOP_2:func.1">-</a> <font color="Maroon" title="c4">c</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="txt" href="complsp1.html#E1:56_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="complsp1.html#T29" target="_self" title="COMPLSP1:th.29">Th29</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">c</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complsp1.html#K15" title="COMPLSP1:func.15">abs</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">j</font>
<b>by </b><i><a class="txt" href="complsp1.html#E1:56_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="complsp1.html#T59" target="_self" title="COMPLSP1:th.59">Th59</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:56"/>
 <a href="complsp1.html#K15" title="COMPLSP1:func.15">abs</a> <span class="p1">(<span class="default"><a href="complsp1.html#K12" title="COMPLSP1:func.12">-</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complsp1.html#K15" title="COMPLSP1:func.15">abs</a> <font color="Maroon" title="c2">z</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T139" title="FINSEQ_2:th.139">FINSEQ_2:139</a></i>;<br/>


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