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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">r</font> be   <a href="real_1.html#NM1">Real</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/><b>let </b><font color="Maroon">x</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/>







<a NAME="E1:79"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">x</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span>
 
<b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E2:79"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z2</font> where <font color="Olive">z2</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font> : <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">z2</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">x</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span>  iff  ex <font color="Olive">z1</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">z1</font> &amp; <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">z1</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">x</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:79"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="complsp1.html#K17" target="_self">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> iff <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="txt" href="complsp1.html#E45"><i><font color="Green">Lemma63</font></i></a></i>;<br/>


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