<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span> holds   <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> (<font color="Olive" title="b2">B</font>,<font color="Olive" title="b1">A</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">A</font>)</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> );<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>,   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>) <span class="kw">-&gt; </span>  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> = <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default">$1 <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> $2</span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">Y</font> = <span class="p1"> { <span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">z1</font>) where <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">z1</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">z1</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> <span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S9" title="DOMAIN_1:sch.9">DOMAIN_1:sch 9</a>();<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> );<br/>
<span class="kw">reconsider </span><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">z1</font>,<font color="Olive" title="b2">z</font>) where <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">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> <span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S9" title="DOMAIN_1:sch.9">DOMAIN_1:sch 9</a>();<br/></span>
<div><span class="lab"><font color="Green" title="E81">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> iff <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )</span><br/><div><span class="lab"><font color="Green" title="E82">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c7">z</font>, <font color="Maroon" title="c8">z1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><a NAME="E2:148_1_1"/><span class="lab"><font color="Green" title="E83">A3</font></span>: 
( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c8">z1</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c8">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">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="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Olive" title="b2">z</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c9">z1</font> = <font color="Maroon" title="c8">z1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="seq_4.html#K14" title="SEQ_4:func.14">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="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z1</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Olive" title="b1">z</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Maroon" title="c9">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c7">z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z1</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c10">z</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Maroon" title="c9">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )</span><br/><span class="kw">thus </span><a NAME="E3:148_1_1"/>
( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z1</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c10">z</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Maroon" title="c9">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_1"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="ref" href="seq_4.html#T104" target="_self" title="SEQ_4:th.104">Th121</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="148_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c7">z1</font>, <font color="Maroon" title="c8">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><a NAME="E2:148_1_2"/><span class="lab"><font color="Green" title="E83">A4</font></span>: 
( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">z1</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c8">z</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Maroon" title="c7">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">z1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">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="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Olive" title="b2">z1</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</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">B</font> &amp; <font color="Olive" title="b2">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c8">z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">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="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Olive" title="b1">z1</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c10">z1</font> = <font color="Maroon" title="c7">z1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c10">z1</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c10">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )</span><br/><span class="kw">thus </span><a NAME="E3:148_1_2"/>
( <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">z</font> <a href="seq_4.html#K19" title="SEQ_4:func.19">-</a> <font color="Maroon" title="c10">z1</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c10">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:148_1_2"><span class="lab"><font color="Green" title="E83">A4</font></span></a>, <a class="ref" href="seq_4.html#T104" target="_self" title="SEQ_4:th.104">Th121</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:148_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> iff <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:148_1"><span class="lab"><font color="Green" title="E82">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:148"/>
(  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</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> &amp;  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">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="c4">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D19" target="_self" title="SEQ_4:def.19">Def22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:148"/>
 <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K26" title="SEQ_4:func.26">dist</a> (<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">A</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="ref" href="subset_1.html#T3" title="SUBSET_1:th.3">SUBSET_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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