<?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#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/>

<b>set </b><font color="Maroon" title="c2">T</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font>;<br/>
<b>thus </b><a NAME="E1:23"/>
( <font color="Maroon" title="c1">M</font> is <a href="tbsp_1.html#V4" title="TBSP_1:attr.4">complete</a> implies for <font color="Olive" title="b1">S</font> being <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="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">S</font> is <a href="kurato_2.html#V1" title="KURATO_2:attr.1">descending</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> not  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E20">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="c3">S</font> be  <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="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">M</font>;<br/>

<b>assume </b><a NAME="E2:23_1"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c3">S</font> is <a href="kurato_2.html#V1" title="KURATO_2:attr.1">descending</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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>$2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E3:23_1"/><i><font color="Green" title="E22">A3</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 <font color="Maroon" title="c1">M</font> &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="23_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c4">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="E2:23_1_1"/><i><font color="Green" title="E24">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">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="E3:23_1_1"/><b>then </b>
not <font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</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#E1:23_1_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:23_1_1"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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/>
<a NAME="E6:23_1_1"/>
( <font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="c3">S</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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 <font color="Maroon" title="c1">M</font> )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E1:23_1_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="compl_sp.html#E2:23_1_1"><i><font color="Green" title="E24">A5</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="E7:23_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">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 <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] )
 <b>by </b><i><a class="txt" href="compl_sp.html#E5:23_1_1"><i><font color="Green" title="E25">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">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> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E5:23_1"/><i><font color="Green" title="E23">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/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">F</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#E3:23_1"><i><font color="Green" title="E22">A3</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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/><a NAME="E1:23_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:23_1_2"/>
<font color="Maroon" title="c4">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E5:23_1"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:23_1"/><b>then </b>
<font color="Maroon" title="c4">F</font> is <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>
 
<b>by </b><i><a class="txt" href="compl_sp.html#E2:23_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="compl_sp.html#T4" target="_self" title="COMPL_SP:th.4">Th4</a></i>;<br/>
<a NAME="E8:23_1"/><b>then </b>
<font color="Maroon" title="c4">F</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#E1:23_1"><i><font color="Green" title="E20">A1</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="c5">x</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="E10:23_1"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="metric_6.html#R1" title="METRIC_6:pred.1">is_convergent_in_metrspace_to</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="metric_6.html#T22" title="METRIC_6:th.22">METRIC_6:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c5">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">F'</font> = <font color="Maroon" title="c4">F</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>consider </b><font color="Maroon" title="c9">F1</font> being    <a href="bhsp_3.html#M1" title="BHSP_3:mode.1">subsequence</a> of <font color="Maroon" title="c7">F'</font><b> such that </b><br/><a NAME="E2:23_1_3"/><i><font color="Green" title="E25">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="frechet2.html#T42" title="FRECHET2:th.42">FRECHET2:42</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">Si</font> = <font color="Maroon" title="c3">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c8">i</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> <font color="Maroon" title="c1">M</font></span>)</span> ;<br/><a NAME="E4:23_1_3"/>
<font color="Maroon" title="c3">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c8">i</font> is <a href="compl_sp.html#V3" title="COMPL_SP:attr.3">closed</a>
 <b>by </b><i><a class="ref" href="compl_sp.html#D8" target="_self" title="COMPL_SP:def.8">def8</a></i>;<br/><a NAME="E5:23_1_3"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c10">Si</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="compl_sp.html#T6" target="_self" title="COMPL_SP:th.6">Th7</a></i>;<br/><a NAME="E6:23_1_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Si</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="E1:23_1_3_1"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c11">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">F1</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:23_1_3_1"/><i><font color="Green" title="E28">A9</font></i>: 
( <font color="Maroon" title="c12">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">F1</font> &amp; <font color="Maroon" title="c9">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:23_1_3_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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#E3:23_1_3_1"><i><font color="Green" title="E28">A9</font></i></a></i>;<br/>
<a NAME="E5:23_1_3_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">i</font> )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E2:23_1_3"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="compl_sp.html#E3:23_1_3_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E6:23_1_3_1"/><b>then </b>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span> &amp; <font color="Maroon" title="c3">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">y</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c8">i</font> )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E2:23_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="compl_sp.html#E5:23_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="kurato_2.html#T22" title="KURATO_2:th.22">KURATO_2:22</a></i>;<br/>
<b>hence </b><a NAME="E7:23_1_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Si</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E7:23_1_3"/><b>then </b><i><font color="Green" title="E27">A8</font></i>: 
 <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c9">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Si</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E5:23_1_3"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="frechet2.html#T12" title="FRECHET2:th.12">FRECHET2:12</a></i>;<br/><a NAME="E8:23_1_3"/>
<font color="Maroon" title="c7">F'</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c6">x'</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E10:23_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="frechet2.html#T31" title="FRECHET2:th.31">FRECHET2:31</a></i>;<br/><a NAME="E9:23_1_3"/><b>then </b>
<font color="Maroon" title="c9">F1</font> <a href="frechet.html#R1" title="FRECHET:pred.1">is_convergent_to</a> <font color="Maroon" title="c6">x'</font>
 <b>by </b><i><a class="ref" href="frechet2.html#T18" title="FRECHET2:th.18">FRECHET2:18</a></i>;<br/><a NAME="E10:23_1_3"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="frechet.html#K1" title="FRECHET:func.1">Lim</a> <font color="Maroon" title="c9">F1</font>
 <b>by </b><i><a class="ref" href="frechet.html#D4" title="FRECHET:def.4">FRECHET:def 4</a></i>;<br/><b>hence </b><a NAME="E11:23_1_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E7:23_1_3"><i><font color="Green" title="E27">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:23_1"/>
not  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c3">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="kurato_2.html#T6" title="KURATO_2:th.6">KURATO_2:6</a></i>;<br/>


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

<b>assume </b><a NAME="E2:23"/><i><font color="Green" title="E20">A1</font></i>: 
for <font color="Olive" title="b1">S</font> being <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="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">S</font> is <a href="kurato_2.html#V1" title="KURATO_2:attr.1">descending</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Olive" title="b1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> not  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Olive" title="b1">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">F</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="E3:23"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">S</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> <a href="compl_sp.html#V7" title="COMPL_SP:attr.7">closed</a> <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E5:23"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">S</font> is <a href="kurato_2.html#V1" title="KURATO_2:attr.1">descending</a>
 <b>and </b><br/><a NAME="E6:23"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c3">F</font> is <a href="tbsp_1.html#V3" title="TBSP_1:attr.3">Cauchy</a> implies ( <font color="Maroon" title="c4">S</font> is <a href="compl_sp.html#V1" title="COMPL_SP:attr.1">bounded</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c4">S</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>and </b><br/><a NAME="E7:23"/><i><font color="Green" title="E24">A5</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">U</font> being  <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> <font color="Maroon" title="c1">M</font></span>)</span> st <br/>( <font color="Olive" title="b2">U</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="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font></span>)</span> where <font color="Olive" title="b3">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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>  &amp; <font color="Maroon" title="c4">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</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> <font color="Olive" title="b2">U</font> )
 <b>by </b><i><a class="ref" href="compl_sp.html#T9" target="_self" title="COMPL_SP:th.9">Th10</a></i>;<br/>
<a NAME="E8:23"/>
not  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c4">S</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#E2:23"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="compl_sp.html#E3:23"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="compl_sp.html#E5:23"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="compl_sp.html#E6:23"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:23"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K2" title="KURATO_2:func.2">meet</a> <font color="Maroon" title="c4">S</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>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="compl_sp.html#E10:23"><i><font color="Green" title="E25">A6</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">x</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="c7">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E12:23"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c8">d</font> =  <a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c4">S</font>;<br/>
<a NAME="E13:23"/>
(  <a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c4">S</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> &amp;  <a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c4">S</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E3:23"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="compl_sp.html#E5:23"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="compl_sp.html#E6:23"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="compl_sp.html#T1" target="_self" title="COMPL_SP:th.1">Th1</a>, <a class="ref" href="compl_sp.html#T2" target="_self" title="COMPL_SP:th.2">Th2</a></i>;<br/>
<a NAME="E14:23"/><b>then </b>
 <a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c4">S</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T52" title="SEQ_4:th.52">SEQ_4:52</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E16:23"/><i><font color="Green" title="E27">A8</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c9">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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:23"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="compl_sp.html#E6:23"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="compl_sp.html#E12:23"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">n</font>
;<br/>

<b>let </b><font color="Maroon" title="c10">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E17:23"/><i><font color="Green" title="E28">A9</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">U</font> being   <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> <font color="Maroon" title="c1">M</font></span>)</span><b> such that </b><br/><a NAME="E19:23"/><i><font color="Green" title="E29">A10</font></i>: 
( <font color="Maroon" title="c11">U</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="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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="c10">m</font> </span>}</span>  &amp; <font color="Maroon" title="c4">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c11">U</font> )
 <b>by </b><i><a class="txt" href="compl_sp.html#E7:23"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/>
<a NAME="E20:23"/>
( <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">U</font> &amp; <font color="Maroon" title="c11">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c11">U</font> )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E19:23"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E21:23"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c10">m</font> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c10">m</font> &amp; <font color="Maroon" title="c4">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c10">m</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#E3:23"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="compl_sp.html#E6:23"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="compl_sp.html#E10:23"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="compl_sp.html#E19:23"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="kurato_2.html#T6" title="KURATO_2:th.6">KURATO_2:6</a>, <a class="ref" href="compl_sp.html#D1" target="_self" title="COMPL_SP:def.1">def1</a></i>;<br/>
<a NAME="E22:23"/><b>then </b>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c10">m</font></span>)</span> &amp;  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="flang_1.html#K5" title="FLANG_1:func.5">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font> )
 
<b>by </b><i><a class="txt" href="compl_sp.html#E16:23"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="compl_sp.html#E17:23"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="tbsp_1.html#D10" title="TBSP_1:def.10">TBSP_1:def 10</a>, <a class="ref" href="tbsp_1.html#T29" title="TBSP_1:th.29">TBSP_1:29</a>, <a class="ref" href="compl_sp.html#D2" target="_self" title="COMPL_SP:def.2">def2</a></i>;<br/>
<a NAME="E23:23"/><b>then </b>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> &amp; <span class="p1">(<span class="default"><a href="compl_sp.html#K1" title="COMPL_SP:func.1">diameter</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">r</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E24:23"/>
not <font color="Maroon" title="c7">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="c3">F</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


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