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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">pmet</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">pmet</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green" title="E16">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds   <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<font color="Olive" title="b1">A</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon" title="c3">PM</font> =  <a href="pcomps_1.html#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font>;<br/>
<a NAME="E3:14"/><i><font color="Green" title="E17">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <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#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span> holds  <font color="Maroon" title="c2">pmet</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <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#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span>;<br/>

<a NAME="E1:14_1"/>
 <a href="pcomps_1.html#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font> #)
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="pcomps_1.html#D8" title="PCOMPS_1:def.8">PCOMPS_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E2:14_1"/>
<font color="Maroon" title="c2">pmet</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:14"/><i><font color="Green" title="E18">A4</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</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="E1:14_2"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">AT</font> = <font color="Maroon" title="c4">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">AP</font> = <font color="Maroon" title="c4">A</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#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span> <b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="nagata_2.html#E1:14_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="pcomps_2.html#T8" title="PCOMPS_2:th.8">PCOMPS_2:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_1"/>
( <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_1"><b>suppose </b></a><a NAME="E1:14_2_1_1"/>
<font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:14_2_1_1"/><b>then </b>
( <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:14_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">AT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T21" title="TOPS_1:th.21">TOPS_1:21</a></i>;<br/><a NAME="E4:14_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">AT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span>
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="pcomps_2.html#T8" title="PCOMPS_2:th.8">PCOMPS_2:8</a></i>;<br/><b>hence </b><a NAME="E5:14_2_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcomps_1.html#T34" title="PCOMPS_1:th.34">PCOMPS_1:34</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1_2"><b>suppose </b></a><a NAME="E1:14_2_1_2"/><i><font color="Green" title="E20">A6</font></i>: 
not <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:14_2_1_2"/>
for <font color="Olive" title="b1">xP</font> being  <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#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span>  st <font color="Olive" title="b1">xP</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">AP</font> holds <br/> ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">xP</font>,<font color="Olive" title="b2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">AP</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">xP</font> be   <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#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:14_2_1_2_1"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c7">xP</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">AP</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">xT</font> = <font color="Maroon" title="c7">xP</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="pcomps_2.html#T8" title="PCOMPS_2:th.8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E3:14_2_1_2_1"/>
<font color="Maroon" title="c5">AT</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14_2"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:14_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T30" title="TOPS_1:th.30">TOPS_1:30</a></i>;<br/>
<a NAME="E5:14_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E6:14_2_1_2_1"/><b>then </b><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:14"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="nagata_2.html#E1:14_2_1_2"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>
<a NAME="E7:14_2_1_2_1"/><i><font color="Green" title="E23">A9</font></i>: 
<span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">xT</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">xT</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:14_2_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">xT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="nagata_2.html#E1:14_2_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="nagata_2.html#T5" target="_self" title="NAGATA_2:th.5">Th5</a>, <a class="ref" href="nagata_2.html#T9" target="_self" title="NAGATA_2:th.9">Th9</a></i>;<br/>
<a NAME="E3:14_2_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c8">xT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c8">xT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AT</font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14_2_1_2_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="nagata_2.html#E6:14_2_1_2_1"><i><font color="Green" title="E22">A8</font></i></a></i>;<br/>
<a NAME="E4:14_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">AT</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E5:14_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b><font color="Maroon" title="c9">r</font> = <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">xT</font>;<br/>

<a NAME="E8:14_2_1_2_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">xP</font>,<font color="Maroon" title="c9">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">AP</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">y</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="E1:14_2_1_2_1_2"/><i><font color="Green" title="E24">A10</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">xP</font>,<font color="Maroon" title="c9">r</font> &amp; not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">AP</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">yP</font> = <font color="Maroon" title="c10">y</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#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">yT</font> = <font color="Maroon" title="c11">yP</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="pcomps_2.html#T8" title="PCOMPS_2:th.8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E4:14_2_1_2_1_2"/><i><font color="Green" title="E25">A11</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">xP</font>,<font color="Maroon" title="c11">yP</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">r</font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14_2_1_2_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E5:14_2_1_2_1_2"/>
( <font color="Maroon" title="c12">yT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c8">xT</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14_2_1_2_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:14_2_1_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c8">xT</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">yT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c8">xT</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Maroon" title="c2">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="nagata_2.html#T9" target="_self" title="NAGATA_2:th.9">Th9</a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:14_2_1_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c8">xT</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">yT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c8">xT</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; not <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c8">xT</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c8">xT</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a> &amp; <font color="Maroon" title="c2">pmet</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14_2_1_2"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="nagata_2.html#E8"><i><font color="Green" title="E8">Lm1</font></i></a></i>;<br/>
<a NAME="E8:14_2_1_2_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c8">xT</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c8">xT</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">yT</font> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c8">xT</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">yT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">xT</font>,<font color="Maroon" title="c12">yT</font> &amp;  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c8">xT</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">xT</font> )
 
<b>by </b><i><a class="ref" href="nagata_2.html#D2" target="_self" title="NAGATA_2:def.2">Def2</a>, <a class="ref" href="nagata_2.html#D3" target="_self" title="NAGATA_2:def.3">Def3</a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:14_2_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="nagata_2.html#E3:14"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="nagata_2.html#E4:14_2_1_2_1_2"><i><font color="Green" title="E25">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:14_2_1_2_1"/>
( <font color="Maroon" title="c9">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">xP</font>,<font color="Maroon" title="c9">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">AP</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E7:14_2_1_2_1"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:14_2_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:14"/>
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</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="E1:14_3"/><i><font color="Green" title="E19">A12</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">AT</font> = <font color="Maroon" title="c4">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="pcomps_2.html#T8" title="PCOMPS_2:th.8">PCOMPS_2:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_3_1"/>
( <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_1_1"><b>suppose </b></a><a NAME="E1:14_3_1_1"/>
<font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:14_3_1_1"/><b>then </b>
( <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:14_3_1_1"/><b>then </b>
<font color="Maroon" title="c5">AT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T21" title="TOPS_1:th.21">TOPS_1:21</a></i>;<br/><b>hence </b><a NAME="E4:14_3_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_3_1_2"><b>suppose </b></a><a NAME="E1:14_3_1_2"/><i><font color="Green" title="E20">A13</font></i>: 
not <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:14_3_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AT</font> iff  ex <font color="Olive" title="b2">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">AT</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:14_3_1_2_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AT</font> implies  ex <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">AT</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_3_1_2_1_1"/><i><font color="Green" title="E21">A14</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AT</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">xT</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">xP</font> = <font color="Maroon" title="c6">x</font> as   <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#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span> <b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="nagata_2.html#E1:14_3_1_2_1_1"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="pcomps_2.html#T8" title="PCOMPS_2:th.8">PCOMPS_2:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:14_3_1_2_1_1"/><i><font color="Green" title="E22">A15</font></i>: 
( <font color="Maroon" title="c9">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">xP</font>,<font color="Maroon" title="c9">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">AT</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:14_3"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="nagata_2.html#E1:14_3_1_2_1_1"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></i>;<br/>
<a NAME="E6:14_3_1_2_1_1"/>
<font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">xT</font>,<font color="Maroon" title="c7">xT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="pcomps_1.html#D7" title="PCOMPS_1:def.7">PCOMPS_1:def 7</a></i>;<br/>
<a NAME="E7:14_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E23">A16</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">xP</font>,<font color="Maroon" title="c8">xP</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">r</font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E3:14"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="nagata_2.html#E5:14_3_1_2_1_1"><i><font color="Green" title="E22">A15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">B</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">xP</font>,<font color="Maroon" title="c9">r</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="pcomps_2.html#T8" title="PCOMPS_2:th.8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E9:14_3_1_2_1_1"/>
( <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp;  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AT</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14_3_1_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="txt" href="nagata_2.html#E5:14_3_1_2_1_1"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:14_3_1_2_1_1"/><i><font color="Green" title="E24">A17</font></i>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">b</font> = <font color="Maroon" title="c11">b</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="nagata_2.html#E11:14_3_1_2_1_1"><i><font color="Green" title="E24">A17</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">Inf</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> ;<br/>
<a NAME="E13:14_3_1_2_1_1"/>
<font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:14_3_1_2_1_1_1"/>
<font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">y</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="E1:14_3_1_2_1_1_1_1"/><i><font color="Green" title="E25">A18</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>

<a NAME="E2:14_3_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="nagata_2.html#E1:14_3_1_2_1_1_1_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="ref" href="nagata_2.html#T6" target="_self" title="NAGATA_2:th.6">Th6</a>, <a class="ref" href="nagata_2.html#T9" target="_self" title="NAGATA_2:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:14_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:14_3_1_2_1_1_1_1"><i><font color="Green" title="E25">A18</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c14">y</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="E2:14_3_1_2_1_1_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> : <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">yT</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:14_3_1_2_1_1_1"/><i><font color="Green" title="E25">A19</font></i>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">yT</font> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K3" title="NAGATA_2:func.3">inf</a> <font color="Maroon" title="c2">pmet</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">yT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">yP</font> = <font color="Maroon" title="c15">yT</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#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span> <b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="pcomps_2.html#T8" title="PCOMPS_2:th.8">PCOMPS_2:8</a></i>;<br/>
<b>assume </b><a NAME="E6:14_3_1_2_1_1_1"/>
not <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>

<a NAME="E7:14_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E26">A20</font></i>: 
<font color="Maroon" title="c15">yT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:14_3_1_2_1_1_1"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:14_3_1_2_1_1_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">xP</font>,<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T33" title="PCOMPS_1:th.33">PCOMPS_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:14_3_1_2_1_1_1"/><i><font color="Green" title="E27">A21</font></i>: 
( <font color="Maroon" title="c17">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c16">yP</font>,<font color="Maroon" title="c17">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c8">xP</font>,<font color="Maroon" title="c9">r</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E7:14_3_1_2_1_1_1"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></i>;<br/>
<a NAME="E11:14_3_1_2_1_1_1"/>
( <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c15">yT</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E11:14_3_1_2_1_1"><i><font color="Green" title="E24">A17</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="E12:14_3_1_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c15">yT</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c15">yT</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Maroon" title="c2">pmet</font> <a href="nagata_1.html#R1" title="NAGATA_1:pred.1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="nagata_2.html#T9" target="_self" title="NAGATA_2:th.9">Th9</a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E13:14_3_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E28">A22</font></i>: 
( not <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c15">yT</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c15">yT</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a> &amp; <font color="Maroon" title="c2">pmet</font> is <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E11:14_3_1_2_1_1"><i><font color="Green" title="E24">A17</font></i></a>, <a class="txt" href="nagata_2.html#E8"><i><font color="Green" title="E8">Lm1</font></i></a></i>;<br/>
<a NAME="E14:14_3_1_2_1_1_1"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c15">yT</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:14_3_1_2_1_1_1"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="nagata_2.html#D3" target="_self" title="NAGATA_2:def.3">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">rn</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E16:14_3_1_2_1_1_1"/><i><font color="Green" title="E29">A23</font></i>: 
( <font color="Maroon" title="c18">rn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c15">yT</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Maroon" title="c18">rn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c17">s</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E10:14_3_1_2_1_1_1"><i><font color="Green" title="E27">A21</font></i></a>, <a class="txt" href="nagata_2.html#E13:14_3_1_2_1_1_1"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:14_3_1_2_1_1_1"/><i><font color="Green" title="E30">A24</font></i>: 
( <font color="Maroon" title="c19">z</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> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c15">yT</font></span>)</span> &amp; <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c18">rn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nagata_2.html#K2" title="NAGATA_2:func.2">dist</a> <font color="Maroon" title="c2">pmet</font>,<font color="Maroon" title="c15">yT</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">z</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E16:14_3_1_2_1_1_1"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">zT</font> = <font color="Maroon" title="c19">z</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="nagata_2.html#E18:14_3_1_2_1_1_1"><i><font color="Green" title="E30">A24</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">zP</font> = <font color="Maroon" title="c19">z</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#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span> <b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="nagata_2.html#E18:14_3_1_2_1_1_1"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="pcomps_2.html#T8" title="PCOMPS_2:th.8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E21:14_3_1_2_1_1_1"/>
<font color="Maroon" title="c2">pmet</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c15">yT</font>,<font color="Maroon" title="c20">zT</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">s</font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E16:14_3_1_2_1_1_1"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="nagata_2.html#E18:14_3_1_2_1_1_1"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="nagata_2.html#D2" target="_self" title="NAGATA_2:def.2">Def2</a></i>;<br/>
<a NAME="E22:14_3_1_2_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c16">yP</font>,<font color="Maroon" title="c21">zP</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">s</font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E3:14"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/>
<a NAME="E23:14_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c21">zP</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c16">yP</font>,<font color="Maroon" title="c17">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E24:14_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">B</font>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E10:14_3_1_2_1_1_1"><i><font color="Green" title="E27">A21</font></i></a>, <a class="txt" href="nagata_2.html#E18:14_3_1_2_1_1_1"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E25:14_3_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:14_3_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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"><font color="Maroon" title="c10">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E2:14"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="nagata_2.html#E11:14_3_1_2_1_1"><i><font color="Green" title="E24">A17</font></i></a></i>;<br/>
<a NAME="E15:14_3_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c10">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">xP</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font> )
 
<b>by </b><i><a class="txt" href="nagata_2.html#E7:14_3_1_2_1_1"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a>, <a class="ref" href="tops_1.html#T30" title="TOPS_1:th.30">TOPS_1:30</a></i>;<br/>
<b>hence </b><a NAME="E16:14_3_1_2_1_1"/>
 ex <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">AT</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> )
 <b>by </b><i><a class="txt" href="nagata_2.html#E5:14_3_1_2_1_1"><i><font color="Green" title="E22">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:14_3_1_2_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">AT</font> iff  ex <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">AT</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> ) )
 ;<br/>


</div><b>end;</b></div><a NAME="E3:14_3_1_2"/><b>then </b>
<font color="Maroon" title="c5">AT</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57" title="TOPS_1:th.57">TOPS_1:57</a></i>;<br/><b>hence </b><a NAME="E4:14_3_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:14"/><b>then </b>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5" title="PCOMPS_1:func.5">SpaceMetr</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">pmet</font></span>)</span>
 
<b>by </b><i><a class="txt" href="nagata_2.html#E4:14"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:14"/>
<font color="Maroon" title="c1">T</font> is <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a>
 <b>by </b><i><a class="txt" href="nagata_2.html#E1:14"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="pcomps_1.html#D9" title="PCOMPS_1:def.9">PCOMPS_1:def 9</a></i>;<br/>


</div>
