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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">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>,   <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>) <b>-&gt; </b>  <a href="real_1.html#NM1">Real</a> = <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">z1</font>,<font color="Olive">z</font>) where <font color="Olive">z1</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>, <font color="Olive">z</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> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z1</font>,<font color="Olive">z</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>from </b><i><a class="ref" href="complsp1.html#S2" target="_self">COMPLSP1:sch 2</a>();<br/></i>
<b>reconsider </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">z</font>,<font color="Olive">z1</font>) where <font color="Olive">z</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>, <font color="Olive">z1</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> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">z</font>,<font color="Olive">z1</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>from </b><i><a class="ref" href="complsp1.html#S2" target="_self">COMPLSP1:sch 2</a>();<br/></i>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">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>, <font color="Maroon">z</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><b> such that </b><a NAME="E1:103_1_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">r</font> <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">z1</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> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><b>take </b><font color="Maroon">z</font> = <font color="Maroon">z</font>;<br/><b>take </b><font color="Maroon">z1</font> = <font color="Maroon">z1</font>;<br/><b>thus </b><a NAME="E2:103_1_1"/>
( <font color="Maroon">r</font> <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">z</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z1</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="complsp1.html#T52" target="_self">Th52</a>, <a class="ref" href="complsp1.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">z</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>, <font color="Maroon">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><b> such that </b><a NAME="E1:103_1_2"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">r</font> <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">z</font> <a href="complsp1.html#K13" target="_self">-</a> <font color="Maroon">z1</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>take </b><font color="Maroon">z1</font> = <font color="Maroon">z1</font>;<br/><b>take </b><font color="Maroon">z</font> = <font color="Maroon">z</font>;<br/><b>thus </b><a NAME="E2:103_1_2"/>
( <font color="Maroon">r</font> <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">z1</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> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="complsp1.html#T52" target="_self">Th52</a>, <a class="ref" href="complsp1.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:103_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="complsp1.html#T51" target="_self">Th51</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:103"/>
(  <a href="complsp1.html#K20" target="_self">dist</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font> &amp;  <a href="complsp1.html#K20" target="_self">dist</a> <font color="Maroon">B</font>,<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="complsp1.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E5:103"/>
 <a href="complsp1.html#K20" target="_self">dist</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="complsp1.html#K20" target="_self">dist</a> <font color="Maroon">B</font>,<font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="complsp1.html#T50" target="_self">Th50</a>, <a class="ref" href="subset_1.html#T8">SUBSET_1:8</a></i>;<br/>


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