<?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#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>assume </b><a NAME="E1:72"/><i><font color="Green" title="E55">A1</font></i>: 
<font color="Maroon" title="c1">M</font> is <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</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>assume </b><a NAME="E2:72"/><i><font color="Green" title="E56">A2</font></i>: 
 ex <font color="Olive" title="b1">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">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="E4:72"/><i><font color="Green" title="E57">A3</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:72"><i><font color="Green" title="E56">A2</font></i></a></i>;<br/>
<a NAME="E5:72"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E4:72"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">X</font> be  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c4">X</font>;<br/>
<b>thus </b><a NAME="E6:72"/>
 <a href="compl_sp.html#K4" title="COMPL_SP:func.4">WellSpace</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">X</font> is <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:72"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="compl_sp.html#T42" target="_self" title="COMPL_SP:th.42">Th42</a></i>;<br/>

<b>set </b><font color="Maroon" title="c6">TW</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <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="c4">X</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c7">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c4">X</font><b> such that </b><br/><a NAME="E8:72"/><i><font color="Green" title="E59">A5</font></i>: 
<font color="Maroon" title="c7">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="dickson.html#T3" title="DICKSON:th.3">DICKSON:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E9:72"/><i><font color="Green" title="E60">A6</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="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></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="72_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:72_1"/><i><font color="Green" title="E61">A7</font></i>: 
<font color="Maroon" title="c8">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>take </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
;<br/>

<a NAME="E2:72_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E1:72_1"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:72_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:72_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>] )
 <b>by </b><i><a class="txt" href="compl_sp.html#E5:72"><i><font color="Green" title="E58">A4</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>
<b>consider </b><font color="Maroon" title="c8">s</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E11:72"/><i><font color="Green" title="E61">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="c8">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#E9:72"><i><font color="Green" title="E60">A6</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E12:72"/><i><font color="Green" title="E62">A9</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">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:72_2"/><i><font color="Green" title="E63">A10</font></i>: 
<font color="Maroon" title="c9">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/>

<a NAME="E2:72_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:72_2"/><b>then </b>
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">s</font>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E1:72_2"><i><font color="Green" title="E63">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:72_2"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">x</font>) is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c4">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T55" title="SUBSET_1:th.55">SUBSET_1:55</a></i>;<br/>
<b>hence </b><a NAME="E5:72_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">S</font> being   <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E14:72"/><i><font color="Green" title="E63">A11</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" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="compl_sp.html#E12:72"><i><font color="Green" title="E62">A9</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:72_3"/><i><font color="Green" title="E64">A12</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">S</font>
 ;<br/><a NAME="E2:72_3"/>
<font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E14:72"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="compl_sp.html#E1:72_3"><i><font color="Green" title="E64">A12</font></i></a></i>;<br/><b>hence </b><a NAME="E3:72_3"/>
not <font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E16:72"/><b>then </b><i><font color="Green" title="E64">A13</font></i>: 
<font color="Maroon" title="c9">S</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">rngs</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">s</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c11">F</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">F</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="relset_2.html#T16" title="RELSET_2:th.16">RELSET_2:16</a></i>;<br/>
<a NAME="E19:72"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:72_4"/><i><font color="Green" title="E65">A14</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">S</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:72_4"/><i><font color="Green" title="E66">A15</font></i>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">S</font> &amp; <font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:72_4"><i><font color="Green" title="E65">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:72_4"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:72_4"/><b>then </b>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E14:72"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="compl_sp.html#E3:72_4"><i><font color="Green" title="E66">A15</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:72_4"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F</font>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c13">R</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E21:72"/><i><font color="Green" title="E65">A16</font></i>: 
<font color="Maroon" title="c13">R</font> is <a href="kurato_2.html#V1" title="KURATO_2:attr.1">descending</a>
 <b>and </b><br/><a NAME="E22:72"/><i><font color="Green" title="E66">A17</font></i>: 
( <font color="Maroon" title="c12">F</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a> &amp; <font color="Maroon" title="c9">S</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c13">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>and </b><br/><a NAME="E23:72"/><i><font color="Green" title="E67">A18</font></i>: 
for <font color="Olive" title="b1">i</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">Si</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c13">R</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">Si</font></span>)</span> &amp; <font color="Olive" title="b2">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b3">j</font></span>)</span> where <font color="Olive" title="b3">j</font> is    <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>  : <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="compl_sp.html#E16:72"><i><font color="Green" title="E64">A13</font></i></a>, <a class="ref" href="compl_sp.html#T23" target="_self" title="COMPL_SP:th.23">Th23</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">R'</font> = <font color="Maroon" title="c13">R</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</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="c4">X</font></span>)</span> ;<br/>
<a NAME="E25:72"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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>by </b><i><a class="txt" href="compl_sp.html#E4:72"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/>
<a NAME="E26:72"/><b>then </b><i><font color="Green" title="E68">A19</font></i>: 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <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>by </b><i><a class="ref" href="xreal_1.html#T131" title="XREAL_1:th.131">XREAL_1:131</a></i>;<br/>
<div><i><font color="Green" title="E69">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">x</font>, <font color="Maroon" title="c16">y</font> be   <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="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span>;<br/><b>assume </b><a NAME="E1:72_5"/><i><font color="Green" title="E70">A21</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">y</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c17">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:72_5"/><i><font color="Green" title="E71">A22</font></i>: 
( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:72_5"><i><font color="Green" title="E70">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:72_5"/><i><font color="Green" title="E72">A23</font></i>: 
( <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:72_5"><i><font color="Green" title="E70">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:72_5"/>
( <font color="Maroon" title="c15">x</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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x1</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c16">y</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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">y1</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E11:72"><i><font color="Green" title="E61">A8</font></i></a>, <a class="txt" href="compl_sp.html#E3:72_5"><i><font color="Green" title="E71">A22</font></i></a>, <a class="txt" href="compl_sp.html#E5:72_5"><i><font color="Green" title="E72">A23</font></i></a></i>;<br/><a NAME="E7:72_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="c4">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</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="c3">b</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#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:72_5"><i><font color="Green" title="E70">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="E8:72_5"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">i</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>consider </b><font color="Maroon" title="c16">Si</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<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="c4">X</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E2:72_6"/><i><font color="Green" title="E70">A24</font></i>: 
<font color="Maroon" title="c13">R</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c16">Si</font></span>)</span>
 <b>and </b><br/><a NAME="E3:72_6"/><i><font color="Green" title="E71">A25</font></i>: 
<font color="Maroon" title="c16">Si</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <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>  : <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="compl_sp.html#E23:72"><i><font color="Green" title="E67">A18</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">SI</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c16">Si</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c4">X</font></span>)</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">x</font>, <font color="Maroon" title="c19">y</font> be   <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="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span>;<br/><b>assume </b><a NAME="E1:72_6_1"/><i><font color="Green" title="E72">A26</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">SI</font> &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">SI</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c20">xS</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:72_6_1"/><i><font color="Green" title="E73">A27</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">xS</font> &amp; <font color="Maroon" title="c20">xS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Si</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:72_6_1"><i><font color="Green" title="E72">A26</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">yS</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:72_6_1"/><i><font color="Green" title="E74">A28</font></i>: 
( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">yS</font> &amp; <font color="Maroon" title="c21">yS</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Si</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:72_6_1"><i><font color="Green" title="E72">A26</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">j1</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="E7:72_6_1"/><i><font color="Green" title="E75">A29</font></i>: 
( <font color="Maroon" title="c20">xS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c22">j1</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:72_6"><i><font color="Green" title="E71">A25</font></i></a>, <a class="txt" href="compl_sp.html#E3:72_6_1"><i><font color="Green" title="E73">A27</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c23">j2</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="E9:72_6_1"/><i><font color="Green" title="E76">A30</font></i>: 
( <font color="Maroon" title="c21">yS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c23">j2</font> &amp; <font color="Maroon" title="c23">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">i</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:72_6"><i><font color="Green" title="E71">A25</font></i></a>, <a class="txt" href="compl_sp.html#E5:72_6_1"><i><font color="Green" title="E74">A28</font></i></a></i>;<br/><a NAME="E10:72_6_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:72_6_1"/><b>then </b>
( <font color="Maroon" title="c8">s</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> &amp; <font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c22">j1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c8">s</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c23">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> &amp; <font color="Maroon" title="c9">S</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c23">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c23">j2</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E14:72"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:72_6_1"/><b>then </b>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> &amp; ( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">y</font> ) )
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:72_6_1"><i><font color="Green" title="E73">A27</font></i></a>, <a class="txt" href="compl_sp.html#E5:72_6_1"><i><font color="Green" title="E74">A28</font></i></a>, <a class="txt" href="compl_sp.html#E7:72_6_1"><i><font color="Green" title="E75">A29</font></i></a>, <a class="txt" href="compl_sp.html#E9:72_6_1"><i><font color="Green" title="E76">A30</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E13:72_6_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c18">x</font>,<font color="Maroon" title="c19">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="compl_sp.html#E26:72"><i><font color="Green" title="E68">A19</font></i></a>, <a class="txt" href="compl_sp.html#E27:72"><i><font color="Green" title="E69">A20</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="E6:72_6"/><b>then </b>
<font color="Maroon" title="c17">SI</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E26:72"><i><font color="Green" title="E68">A19</font></i></a>, <a class="ref" href="tbsp_1.html#D9" title="TBSP_1:def.9">TBSP_1:def 9</a></i>;<br/><b>hence </b><a NAME="E7:72_6"/>
<font color="Maroon" title="c14">R'</font> <a href="compl_sp.html#K2" title="COMPL_SP:func.2">.</a> <font color="Maroon" title="c15">i</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E2:72_6"><i><font color="Green" title="E70">A24</font></i></a>, <a class="ref" href="compl_sp.html#T8" target="_self" title="COMPL_SP:th.8">Th8</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c15">R'</font> = <font color="Maroon" title="c14">R'</font> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</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="c4">X</font></span>)</span> <b>by </b><i><a class="ref" href="compl_sp.html#D1" target="_self" title="COMPL_SP:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c15">R'</font>
;<br/>

<b>thus </b><a NAME="E30:72"/>
( <font color="Maroon" title="c15">R'</font> is <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> &amp; <font color="Maroon" title="c15">R'</font> is <a href="kurato_2.html#V1" title="KURATO_2:attr.1">descending</a> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E21:72"><i><font color="Green" title="E65">A16</font></i></a>, <a class="ref" href="compl_sp.html#T7" target="_self" title="COMPL_SP:th.7">Th7</a></i>;<br/>

<a NAME="E31:72"/><i><font color="Green" title="E70">A31</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="c2">a</font>,<font color="Maroon" title="c4">X</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">rngs</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E27:72"><i><font color="Green" title="E69">A20</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">x1</font>, <font color="Maroon" title="c17">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:72_7"/><i><font color="Green" title="E71">A32</font></i>: 
( <font color="Maroon" title="c16">x1</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="c17">x2</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">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> )
 ;<br/><a NAME="E2:72_7"/>
( <font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c9">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E14:72"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="compl_sp.html#E1:72_7"><i><font color="Green" title="E71">A32</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:72_7"/><b>then </b>
( <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> &amp; <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font></span>)</span>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E11:72"><i><font color="Green" title="E61">A8</font></i></a>, <a class="txt" href="compl_sp.html#E1:72_7"><i><font color="Green" title="E71">A32</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:72_7"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E5:72_7"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E8:72"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="compl_sp.html#E1:72_7"><i><font color="Green" title="E71">A32</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E33:72"/>
 <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c15">R'</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E22:72"><i><font color="Green" title="E66">A17</font></i></a>, <a class="txt" href="compl_sp.html#E26:72"><i><font color="Green" title="E68">A19</font></i></a>, <a class="txt" href="compl_sp.html#E31:72"><i><font color="Green" title="E70">A31</font></i></a>, <a class="txt" href="compl_sp.html#E47"><i><font color="Green" title="E43">Lm7</font></i></a>, <a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>;<br/>


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