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

<b>let </b><font color="Maroon" title="c1">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#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="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  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>consider </b><font color="Maroon" title="c4">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:69"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c4">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E3:69"/><i><font color="Green" title="E53">A2</font></i>: 
 <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="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">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>let </b><font color="Maroon" title="c6">S</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tbsp_1.html#D6" title="TBSP_1:def.6">TBSP_1:def 6</a><br/>

<b>assume </b><a NAME="E4:69"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c6">S</font> is <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( <font color="Maroon" title="c6">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> implies $2 <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="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( <font color="Maroon" title="c6">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> implies $2 <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="c3">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) );<br/>
<a NAME="E5:69"/><i><font color="Green" title="E55">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:69_1"/><i><font color="Green" title="E56">A5</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">i</font> = <font color="Maroon" title="c7">x</font> as    <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>by </b><i><a class="txt" href="compl_sp.html#E1:69_1"><i><font color="Green" title="E56">A5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_1"/>
( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c8">i</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="c6">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1"/><i><font color="Green" title="E57">A6</font></i>: 
<font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c9">xS</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><b>thus </b><a NAME="E2:69_1_1_1"/>
( <font color="Maroon" title="c9">xS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">xS</font>] )
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:69"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="compl_sp.html#E1:69_1_1_1"><i><font color="Green" title="E57">A6</font></i></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_2"/><i><font color="Green" title="E57">A7</font></i>: 
<font color="Maroon" title="c6">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c9">xS</font> = <span class="p1"><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>;<br/><b>thus </b><a NAME="E2:69_1_1_2"/>
( <font color="Maroon" title="c9">xS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">xS</font>] )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:69_1_1_2"><i><font color="Green" title="E57">A7</font></i></a>, <a class="ref" href="compl_sp.html#T38" target="_self" title="COMPL_SP:th.38">Th38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">S'</font> being   <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="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span><b> such that </b><br/><a NAME="E7:69"/><i><font color="Green" title="E56">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">S'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="compl_sp.html#E5:69"><i><font color="Green" title="E55">A4</font></i></a>);<br/></i>
<a NAME="E8:69"/>
<font color="Maroon" title="c7">S'</font> is <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="tbsp_1.html#D5" title="TBSP_1:def.5">TBSP_1:def 5</a><br/>

<b>assume </b><a NAME="E1:69_2"/><i><font color="Green" title="E57">A9</font></i>: 
<font color="Maroon" title="c8">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> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">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="E3:69_2"/><i><font color="Green" title="E58">A10</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="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">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="c6">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="c8">r</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E4:69"><i><font color="Green" title="E54">A3</font></i></a>, <a class="txt" href="compl_sp.html#E1:69_2"><i><font color="Green" title="E57">A9</font></i></a>, <a class="ref" href="tbsp_1.html#D5" title="TBSP_1:def.5">TBSP_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">p</font>
;<br/>

<b>let </b><font color="Maroon" title="c10">n</font>, <font color="Maroon" title="c11">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E4:69_2"/><i><font color="Green" title="E59">A11</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c9">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_2_1"/>
( ( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) or ( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</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="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) or ( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</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="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</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="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1"><b>suppose </b></a><a NAME="E1:69_2_1_1"/>
( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 ;<br/><div class="add"><a NAME="E2:69_2_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E7:69"><i><font color="Green" title="E56">A8</font></i></a></i>;<br/><a NAME="E3:69_2_1_1"/><b>then </b>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E4:69_2_1_1"/>
not <font color="Maroon" title="c8">r</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:69_2"><i><font color="Green" title="E58">A10</font></i></a>, <a class="txt" href="compl_sp.html#E4:69_2"><i><font color="Green" title="E59">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_2"><b>suppose </b></a><a NAME="E1:69_2_1_2"/><i><font color="Green" title="E60">A12</font></i>: 
( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</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="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 ;<br/><div class="add"><a NAME="E2:69_2_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font> &amp; <font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:69"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="compl_sp.html#E7:69"><i><font color="Green" title="E56">A8</font></i></a></i>;<br/><a NAME="E3:69_2_1_2"/><b>then </b>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <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#E1:69_2_1_2"><i><font color="Green" title="E60">A12</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#T1" title="METRIC_1:th.1">METRIC_1:1</a></i>;<br/><b>hence </b><a NAME="E4:69_2_1_2"/>
not <font color="Maroon" title="c8">r</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:69_2"><i><font color="Green" title="E58">A10</font></i></a>, <a class="txt" href="compl_sp.html#E4:69_2"><i><font color="Green" title="E59">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_3"><b>suppose </b></a><a NAME="E1:69_2_1_3"/><i><font color="Green" title="E60">A13</font></i>: 
( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> )
 ;<br/><div class="add"><a NAME="E2:69_2_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font> &amp; <font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:69"><i><font color="Green" title="E52">A1</font></i></a>, <a class="txt" href="compl_sp.html#E7:69"><i><font color="Green" title="E56">A8</font></i></a></i>;<br/><a NAME="E3:69_2_1_3"/><b>then </b>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span>)</span> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <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#E1:69_2_1_3"><i><font color="Green" title="E60">A13</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#T1" title="METRIC_1:th.1">METRIC_1:1</a></i>;<br/><b>hence </b><a NAME="E4:69_2_1_3"/>
not <font color="Maroon" title="c8">r</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:69_2"><i><font color="Green" title="E58">A10</font></i></a>, <a class="txt" href="compl_sp.html#E4:69_2"><i><font color="Green" title="E59">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_4"><b>suppose </b></a><a NAME="E1:69_2_1_4"/>
( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</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="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> )
 ;<br/><div class="add"><a NAME="E2:69_2_1_4"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E7:69"><i><font color="Green" title="E56">A8</font></i></a></i>;<br/><a NAME="E3:69_2_1_4"/><b>then </b>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="compl_sp.html#D10" target="_self" title="COMPL_SP:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E4:69_2_1_4"/>
not <font color="Maroon" title="c8">r</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:69_2"><i><font color="Green" title="E58">A10</font></i></a>, <a class="txt" href="compl_sp.html#E4:69_2"><i><font color="Green" title="E59">A11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:69"/><b>then </b>
<font color="Maroon" title="c7">S'</font> is <a href="tbsp_1.html#V2" title="TBSP_1:attr.2">convergent</a>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E3:69"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="tbsp_1.html#D6" title="TBSP_1:def.6">TBSP_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">L</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c2">a</font>,<font color="Maroon" title="c3">X</font></span>)</span><b> such that </b><br/><a NAME="E11:69"/><i><font color="Green" title="E57">A14</font></i>: 
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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">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="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c8">L</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="ref" href="tbsp_1.html#D3" title="TBSP_1:def.3">TBSP_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">L1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c10">L2</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="E13:69"/><i><font color="Green" title="E58">A15</font></i>: 
( <font color="Maroon" title="c8">L</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">L1</font>,<font color="Maroon" title="c10">L2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; ( ( <font color="Maroon" title="c9">L1</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">L2</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">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c10">L2</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">Th38</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">L2</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="tbsp_1.html#D3" title="TBSP_1:def.3">TBSP_1:def 3</a><br/>

<b>let </b><font color="Maroon" title="c11">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E14:69"/><i><font color="Green" title="E59">A16</font></i>: 
<font color="Maroon" title="c11">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> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c12">n</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="E16:69"/><i><font color="Green" title="E60">A17</font></i>: 
for <font color="Olive" title="b1">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="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span>,<font color="Maroon" title="c8">L</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">r</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E11:69"><i><font color="Green" title="E57">A14</font></i></a>, <a class="txt" href="compl_sp.html#E14:69"><i><font color="Green" title="E59">A16</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">n</font>
;<br/>

<b>let </b><font color="Maroon" title="c13">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E17:69"/><i><font color="Green" title="E61">A18</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_3"/>
( ( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c9">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> ) or ( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c9">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">X</font> ) or ( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</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">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font> ) or ( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</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">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_1"><b>suppose </b></a><a NAME="E1:69_3_1"/><i><font color="Green" title="E62">A19</font></i>: 
( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c9">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> )
 ;<br/><div class="add"><a NAME="E2:69_3_1"/><b>then </b>
<font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">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="c3">X</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E7:69"><i><font color="Green" title="E56">A8</font></i></a></i>;<br/><a NAME="E3:69_3_1"/><b>then </b>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c10">L2</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E13:69"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="compl_sp.html#E1:69_3_1"><i><font color="Green" title="E62">A19</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="E4:69_3_1"/>
not <font color="Maroon" title="c11">r</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c10">L2</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E16:69"><i><font color="Green" title="E60">A17</font></i></a>, <a class="txt" href="compl_sp.html#E17:69"><i><font color="Green" title="E61">A18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_2"><b>suppose </b></a><a NAME="E1:69_3_2"/><i><font color="Green" title="E62">A20</font></i>: 
( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c9">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">X</font> )
 ;<br/><div class="add"><a NAME="E2:69_3_2"/><b>then </b>
<font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">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="c3">X</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E7:69"><i><font color="Green" title="E56">A8</font></i></a></i>;<br/><a NAME="E3:69_3_2"/><b>then </b>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c8">L</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c10">L2</font></span>)</span> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span> <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#E13:69"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="compl_sp.html#E1:69_3_2"><i><font color="Green" title="E62">A20</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#T1" title="METRIC_1:th.1">METRIC_1:1</a></i>;<br/><b>hence </b><a NAME="E4:69_3_2"/>
not <font color="Maroon" title="c11">r</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c10">L2</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E16:69"><i><font color="Green" title="E60">A17</font></i></a>, <a class="txt" href="compl_sp.html#E17:69"><i><font color="Green" title="E61">A18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_3"><b>suppose </b></a><a NAME="E1:69_3_3"/><i><font color="Green" title="E62">A21</font></i>: 
( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</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">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x0</font> )
 ;<br/><div class="add"><a NAME="E2:69_3_3"/><b>then </b>
<font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">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="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E7:69"><i><font color="Green" title="E56">A8</font></i></a></i>;<br/><a NAME="E3:69_3_3"/><b>then </b>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c8">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c10">L2</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E13:69"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="compl_sp.html#E1:69_3_3"><i><font color="Green" title="E62">A21</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="E4:69_3_3"/>
not <font color="Maroon" title="c11">r</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c10">L2</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E16:69"><i><font color="Green" title="E60">A17</font></i></a>, <a class="txt" href="compl_sp.html#E17:69"><i><font color="Green" title="E61">A18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_4"><b>suppose </b></a><a NAME="E1:69_3_4"/><i><font color="Green" title="E62">A22</font></i>: 
( <font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</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">L1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x0</font> )
 ;<br/><div class="add"><a NAME="E2:69_3_4"/><b>then </b>
<font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">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="c4">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E7:69"><i><font color="Green" title="E56">A8</font></i></a></i>;<br/><a NAME="E3:69_3_4"/><b>then </b>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c8">L</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> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<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="c10">L2</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<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="c10">L2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c10">L2</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c8">L</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">r</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E13:69"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="compl_sp.html#E16:69"><i><font color="Green" title="E60">A17</font></i></a>, <a class="txt" href="compl_sp.html#E17:69"><i><font color="Green" title="E61">A18</font></i></a>, <a class="txt" href="compl_sp.html#E1:69_3_4"><i><font color="Green" title="E62">A22</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="E4:69_3_4"/>
not <font color="Maroon" title="c11">r</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">m</font></span>)</span>,<font color="Maroon" title="c10">L2</font>
 <b>by </b><i><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><b>end;</b></div>

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