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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">x</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/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" title="COMPLSP1:func.8">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:87"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">z1</font> being    <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><b> such that </b><br/><a NAME="E3:87"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c4">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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>) <b>-&gt; </b>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> = <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> $1</span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>) where <font color="Olive" title="b1">z</font> is    <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> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>from </b><i><a class="ref" href="complsp1.html#S1" target="_self" title="COMPLSP1:sch.1">COMPLSP1:sch 1</a>();<br/></i>
<a NAME="E5:87"/><i><font color="Green" title="E54">A2</font></i>: 
 <a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c5">X</font>
 
<b>by </b><i><a class="ref" href="complsp1.html#D18" target="_self" title="COMPLSP1:def.18">Def18</a></i>;<br/>
<a NAME="E6:87"/><i><font color="Green" title="E55">A3</font></i>: 
<span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Maroon" title="c4">z1</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 
<b>by </b><i><a class="txt" href="complsp1.html#E3:87"><i><font color="Green" title="E53">A1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">r'</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:87_1"/>
<font color="Maroon" title="c6">r'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 ;<br/><a NAME="E2:87_1"/><b>then </b>
 ex <font color="Olive" title="b1">z</font> being   <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> st <br/>( <font color="Maroon" title="c6">r'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complsp1.html#K16" title="COMPLSP1:func.16">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="complsp1.html#K13" title="COMPLSP1:func.13">-</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="complsp1.html#K16" title="COMPLSP1:func.16">.|</a></span> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 ;<br/><b>hence </b><a NAME="E3:87_1"/>
<font color="Maroon" title="c6">r'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="complsp1.html#T65" target="_self" title="COMPLSP1:th.65">Th65</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:87"/>
 <a href="complsp1.html#K18" title="COMPLSP1:func.18">dist</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="complsp1.html#E5:87"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="complsp1.html#E6:87"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="complsp1.html#T84" target="_self" title="COMPLSP1:th.84">Th84</a></i>;<br/>


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