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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">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1" target="_self">REAL</a> <font color="Maroon">n</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_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="E1:18_1"/>
<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>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>, <font color="Maroon">r'</font> = <span class="p1">(<span class="default"><a href="euclid.html#K6" target="_self">-</a> <font color="Maroon">x</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#K1">REAL</a>  <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="euclid.html#K10" target="_self">abs</a> <span class="p2">(<span class="default"><a href="euclid.html#K6" target="_self">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r'</font>
<b>by </b><i/>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T35">RVSUM_1:35</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K10" target="_self">abs</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18"/>
 <a href="euclid.html#K10" target="_self">abs</a> <span class="p1">(<span class="default"><a href="euclid.html#K6" target="_self">-</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K10" target="_self">abs</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>;<br/>


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