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

<b>let </b><font color="Maroon" title="c1">M</font> be    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>set </b><font color="Maroon" title="c4">XX</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">w</font> =  <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font>;<br/>
<b>set </b><font color="Maroon" title="c6">W</font> =  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font>;<br/>
<b>thus </b><a NAME="E1:64"/>
( <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> implies  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font> is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">A</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c9">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E2:64_1_1"/><i><font color="Green" title="E54">A2</font></i>: 
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) or ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) ) )
 <b>by </b><i><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th39</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font> = 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="txt" href="compl_sp.html#E2:64_1_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="compl_sp.html#E1:64_1"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E3:64_1"/><b>then </b>
 <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font> is <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D3" title="METRIC_1:def.3">METRIC_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:64_1"/>
 <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font> is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D7" title="METRIC_1:def.7">METRIC_1:def 7</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:64"/>
( <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> implies  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_2"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>consider </b><font color="Maroon" title="c9">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c10">b1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E2:64_2_1"/><i><font color="Green" title="E54">A2</font></i>: 
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y1</font>,<font color="Maroon" title="c10">b1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( ( <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c10">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) or ( <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) ) )
 <b>by </b><i><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th39</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c12">b2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E4:64_2_1"/><i><font color="Green" title="E55">A3</font></i>: 
( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( ( <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c12">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) or ( <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) ) )
 <b>by </b><i><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th39</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_2_1_1_1"/>
( ( <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> ) or <font color="Maroon" title="c9">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y2</font> or ( <font color="Maroon" title="c10">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c12">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font> ) )
 </span><b>by </b><i><a class="txt" href="compl_sp.html#E2:64_2_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="compl_sp.html#E4:64_2_1"><i><font color="Green" title="E55">A3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:64_2_1_1_1_1"/>
( <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:64_2_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="compl_sp.html#E4:64_2_1"><i><font color="Green" title="E55">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:64_2_1_1_1_2"/><i><font color="Green" title="E56">A4</font></i>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y2</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> = 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="compl_sp.html#E2:64_2_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="compl_sp.html#E4:64_2_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">b1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="compl_sp.html#E1:64_2"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="metric_1.html#T3" title="METRIC_1:th.3">METRIC_1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">b1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c12">b2</font>,<font color="Maroon" title="c2">a</font></span>)</span>
<b>by </b><i><a class="txt" href="compl_sp.html#E1:64_2"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="metric_1.html#T3" title="METRIC_1:th.3">METRIC_1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="txt" href="compl_sp.html#E2:64_2_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="compl_sp.html#E4:64_2_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="compl_sp.html#E1:64_2_1_1_1_2"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:64_2_1_1_1_3"/><i><font color="Green" title="E56">A4</font></i>: 
( <font color="Maroon" title="c10">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c12">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font> )
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> = 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c12">b2</font>
<b>by </b><i><a class="txt" href="compl_sp.html#E2:64_2_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="compl_sp.html#E4:64_2_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a></i>
<br/>.= 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c12">b2</font>,<font color="Maroon" title="c10">b1</font>
<b>by </b><i><a class="txt" href="compl_sp.html#E1:64_2"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="metric_1.html#T3" title="METRIC_1:th.3">METRIC_1:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c7">A</font>
<b>by </b><i><a class="txt" href="compl_sp.html#E2:64_2_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="compl_sp.html#E4:64_2_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="compl_sp.html#E1:64_2_1_1_1_3"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:64_2_1"/>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c7">A</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E3:64_2"/><b>then </b>
 <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D5" title="METRIC_1:def.5">METRIC_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:64_2"/>
 <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D9" title="METRIC_1:def.9">METRIC_1:def 9</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:64"/>
( <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> implies  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_3"/><i><font color="Green" title="E53">A1</font></i>: 
( <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font>, <font color="Maroon" title="c9">C</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>consider </b><font color="Maroon" title="c10">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c11">b1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E2:64_3_1"/><i><font color="Green" title="E54">A2</font></i>: 
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y1</font>,<font color="Maroon" title="c11">b1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( ( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c11">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) or ( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c11">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) ) )
 <b>by </b><i><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th39</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c13">b2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E4:64_3_1"/><i><font color="Green" title="E55">A3</font></i>: 
( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c13">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( ( <font color="Maroon" title="c12">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c13">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) or ( <font color="Maroon" title="c12">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c13">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) ) )
 <b>by </b><i><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th39</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">y3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c15">b3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E6:64_3_1"/><i><font color="Green" title="E56">A4</font></i>: 
( <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y3</font>,<font color="Maroon" title="c15">b3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( ( <font color="Maroon" title="c14">y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c15">b3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) or ( <font color="Maroon" title="c14">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c15">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) ) )
 <b>by </b><i><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th39</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_3_1_1_1"/>
( ( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y3</font> ) or ( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y3</font> ) or ( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">y3</font> ) or ( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">y3</font> &amp; <font color="Maroon" title="c12">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">y3</font> ) or ( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">y3</font> &amp; <font color="Maroon" title="c12">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y3</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:64_3_1_1_1_1"/>
( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y3</font> )
 ;<br/><div class="add"><a NAME="E2:64_3_1_1_1_1"/><b>then </b>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c15">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c13">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c15">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c15">b3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c13">b2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c15">b3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:64_3"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="compl_sp.html#E2:64_3_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="compl_sp.html#E4:64_3_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="compl_sp.html#E6:64_3_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/><b>hence </b><a NAME="E3:64_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:64_3_1_1_1_2"/>
( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y3</font> )
 ;<br/><div class="add"><a NAME="E2:64_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E57">A5</font></i>: 
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c15">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c13">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c15">b3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c13">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c13">b2</font></span>)</span> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c15">b3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c15">b3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:64_3"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="compl_sp.html#E2:64_3_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="compl_sp.html#E4:64_3_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="compl_sp.html#E6:64_3_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/><a NAME="E3:64_3_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c13">b2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c15">b3</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font></span>)</span> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c15">b3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c13">b2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c15">b3</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="txt" href="compl_sp.html#E1:64_3"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/><b>hence </b><a NAME="E4:64_3_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:64_3_1_1_1_2"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1_1_3"><b>suppose </b></a><a NAME="E1:64_3_1_1_1_3"/>
( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">y3</font> )
 ;<br/><div class="add"><a NAME="E2:64_3_1_1_1_3"/><b>then </b><i><font color="Green" title="E57">A5</font></i>: 
( <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c15">b3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c13">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c15">b3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c13">b2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c2">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:64_3"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="compl_sp.html#E2:64_3_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="compl_sp.html#E4:64_3_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="compl_sp.html#E6:64_3_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/><a NAME="E3:64_3_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c13">b2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c15">b3</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E4:64_3_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:64_3_1_1_1_3"><i><font color="Green" title="E57">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1_1_4"><b>suppose </b></a><a NAME="E1:64_3_1_1_1_4"/>
( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">y3</font> &amp; <font color="Maroon" title="c12">y2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">y3</font> )
 ;<br/><div class="add"><a NAME="E2:64_3_1_1_1_4"/><b>then </b><i><font color="Green" title="E57">A5</font></i>: 
( <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c15">b3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c13">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c15">b3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:64_3_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="compl_sp.html#E4:64_3_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="compl_sp.html#E6:64_3_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a></i>;<br/><a NAME="E3:64_3_1_1_1_4"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c13">b2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:64_3"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/><a NAME="E4:64_3_1_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c15">b3</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:64_3_1_1_1_4"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E5:64_3_1_1_1_4"/>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:64_3_1_1_1_4"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1_1_5"><b>suppose </b></a><a NAME="E1:64_3_1_1_1_5"/>
( <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y2</font> &amp; <font color="Maroon" title="c10">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">y3</font> &amp; <font color="Maroon" title="c12">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y3</font> )
 ;<br/><div class="add"><a NAME="E2:64_3_1_1_1_5"/><b>then </b><i><font color="Green" title="E57">A5</font></i>: 
( <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c15">b3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c13">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c15">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c15">b3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c13">b2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c15">b3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:64_3"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="compl_sp.html#E2:64_3_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="compl_sp.html#E4:64_3_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="compl_sp.html#E6:64_3_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/><a NAME="E3:64_3_1_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c11">b1</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c13">b2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c13">b2</font>,<font color="Maroon" title="c15">b3</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:64_3_1_1_1_5"/>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:64_3_1_1_1_5"><i><font color="Green" title="E57">A5</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:64_3_1"/>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">B</font>,<font color="Maroon" title="c9">C</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E3:64_3"/><b>then </b>
 <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font> is <a href="metric_1.html#V5" title="METRIC_1:attr.5">triangle</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D6" title="METRIC_1:def.6">METRIC_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:64_3"/>
 <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D10" title="METRIC_1:def.10">METRIC_1:def 10</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:64"/><i><font color="Green" title="E53">A1</font></i>: 
( <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> &amp; <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><b>assume </b><a NAME="E1:64_4"/><i><font color="Green" title="E54">A2</font></i>: 
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c9">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c10">b1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E3:64_4"/><i><font color="Green" title="E55">A3</font></i>: 
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y1</font>,<font color="Maroon" title="c10">b1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( ( <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c10">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) or ( <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) ) )
 <b>by </b><i><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th39</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c12">b2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E5:64_4"/><i><font color="Green" title="E56">A4</font></i>: 
( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( ( <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c12">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) or ( <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) ) )
 <b>by </b><i><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th39</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_4_1_1"/>
( <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font> or <font color="Maroon" title="c9">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1_1_1"><b>suppose </b></a><a NAME="E1:64_4_1_1_1"/><i><font color="Green" title="E57">A5</font></i>: 
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y2</font>
 ;<br/><div class="add"><a NAME="E2:64_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c12">b2</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:64_4"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="compl_sp.html#E5:64_4"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a></i>;<br/><b>hence </b><a NAME="E3:64_4_1_1_1"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E4:64"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="compl_sp.html#E1:64_4"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a>, <a class="txt" href="compl_sp.html#E3:64_4"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="compl_sp.html#E5:64_4"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="compl_sp.html#E1:64_4_1_1_1"><i><font color="Green" title="E57">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1_1_2"><b>suppose </b></a><a NAME="E1:64_4_1_1_2"/>
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y2</font>
 ;<br/><div class="add"><a NAME="E2:64_4_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">b2</font></span>)</span> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">b2</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>  &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c2">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="compl_sp.html#E4:64"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="compl_sp.html#E3:64_4"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="compl_sp.html#E5:64_4"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">def10</a>, <a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/><a NAME="E3:64_4_1_1_2"/><b>then </b>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c12">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:64_4"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></i>;<br/><b>hence </b><a NAME="E4:64_4_1_1_2"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E4:64"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a>, <a class="txt" href="compl_sp.html#E3:64_4"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="compl_sp.html#E5:64_4"><i><font color="Green" title="E56">A4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:64_4"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E6:64"/><b>then </b>
 <a href="compl_sp.html#K3" title="COMPL_SP:func.3">well_dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font> is <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D4" title="METRIC_1:def.4">METRIC_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:64"/>
 <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font> is <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D8" title="METRIC_1:def.8">METRIC_1:def 8</a></i>;<br/>


</div>
