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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font>;<br/>





<b>set </b><font color="Maroon" title="c4">W</font> =  <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>let </b><font color="Maroon" title="c5">S</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:70"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">Xa</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1"/>
( for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b3">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">n</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span>,<font color="Maroon" title="c6">Xa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> or  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b2">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b3">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">n</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span>,<font color="Maroon" title="c6">Xa</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font> ) ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1"><b>suppose </b></a><a NAME="E1:70_1_1"/>
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b3">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">n</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span>,<font color="Maroon" title="c6">Xa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:70_1_1"/>
( for <font color="Olive" title="b1">Xa</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">Xa</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="c1">X</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b3">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b4">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b4">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">n</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b4">m</font></span>)</span>,<font color="Olive" title="b1">Xa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> or  ex <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ex <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>for <font color="Olive" title="b3">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b4">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font> st <font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b3">m</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="Olive" title="b2">Y</font>,<font color="Olive" title="b4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2"><b>suppose </b></a><a NAME="E1:70_1_2"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b2">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b3">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">n</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b3">m</font></span>)</span>,<font color="Maroon" title="c6">Xa</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">r</font> ) ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:70_1_2"/><i><font color="Green" title="E54">A2</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E4:70_1_2"/><i><font color="Green" title="E55">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   ex <font color="Olive" title="b2">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span>,<font color="Maroon" title="c6">Xa</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">r</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c8">p</font> being    <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> <b> such that </b><br/><a NAME="E6:70_1_2"/><i><font color="Green" title="E56">A4</font></i>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:70"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="compl_sp.html#E3:70_1_2"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="tbsp_1.html#D5" title="TBSP_1:def.5">TBSP_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">p'</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E8:70_1_2"/><i><font color="Green" title="E57">A5</font></i>: 
( <font color="Maroon" title="c9">p'</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c9">p'</font></span>)</span>,<font color="Maroon" title="c6">Xa</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">r</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E4:70_1_2"><i><font color="Green" title="E55">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c11">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E10:70_1_2"/><i><font color="Green" title="E58">A6</font></i>: 
( <font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c9">p'</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">Y</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( ( <font color="Maroon" title="c10">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font> ) or ( <font color="Maroon" title="c10">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> ) ) )
 <b>by </b><i><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></i>;<br/><a NAME="E11:70_1_2"/>
 ex <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ex <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>for <font color="Olive" title="b3">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b4">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font> st <font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b3">m</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="Olive" title="b2">Y</font>,<font color="Olive" title="b4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c9">p'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c10">Y</font>
;<br/>

<b>let </b><font color="Maroon" title="c12">m</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:70_1_2_1"/><i><font color="Green" title="E59">A7</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">p'</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c14">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E3:70_1_2_1"/><i><font color="Green" title="E60">A8</font></i>: 
( <font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c12">m</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="c13">Z</font>,<font color="Maroon" title="c14">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( ( <font color="Maroon" title="c13">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c14">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font> ) or ( <font color="Maroon" title="c13">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> ) ) )
 <b>by </b><i><a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></i>;<br/>
<a NAME="E4:70_1_2_1"/>
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1_2_1_1"/><i><font color="Green" title="E61">A9</font></i>: 
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">Z</font>
 ;<br/>

<a NAME="E2:70_1_2_1_1"/>
( ( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Y</font> or <font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">Y</font> ) &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</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="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></i>;<br/>
<a NAME="E3:70_1_2_1_1"/><b>then </b>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c9">p'</font></span>)</span>,<font color="Maroon" title="c6">Xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c3">a</font> or  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c9">p'</font></span>)</span>,<font color="Maroon" title="c6">Xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c3">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>  )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E10:70_1_2"><i><font color="Green" title="E58">A6</font></i></a>, <a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></i>;<br/>
<a NAME="E4:70_1_2_1_1"/><b>then </b>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">r</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c14">z</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#E8:70_1_2"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/>
<a NAME="E5:70_1_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c3">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#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c14">z</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">r</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c9">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E8:70_1_2"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="compl_sp.html#E1:70_1_2_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:70_1_2_1_1"/><b>then </b>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c9">p'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">r</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c9">p'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c12">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font> )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E6:70_1_2"><i><font color="Green" title="E56">A4</font></i></a>, <a class="txt" href="compl_sp.html#E8:70_1_2"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="compl_sp.html#E10:70_1_2"><i><font color="Green" title="E58">A6</font></i></a>, <a class="txt" href="compl_sp.html#E3:70_1_2_1"><i><font color="Green" title="E60">A8</font></i></a>, <a class="txt" href="compl_sp.html#E1:70_1_2_1_1"><i><font color="Green" title="E61">A9</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="E7:70_1_2_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:70_1_2_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font> st <font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c12">m</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">Y</font>,<font color="Olive" title="b1">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:70_1_2_1"><i><font color="Green" title="E60">A8</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:70_1_2"/>
( for <font color="Olive" title="b1">Xa</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c1">X</font></span>)</span>  st <font color="Olive" title="b1">Xa</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="c1">X</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b3">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>for <font color="Olive" title="b4">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b4">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">n</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b4">m</font></span>)</span>,<font color="Olive" title="b1">Xa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> or  ex <font color="Olive" title="b1">n</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ex <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>for <font color="Olive" title="b3">m</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b4">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font> st <font color="Maroon" title="c5">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b3">m</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="Olive" title="b2">Y</font>,<font color="Olive" title="b4">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
