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

<b>let </b><font color="Maroon">PM</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/>

<b>set </b><font color="Maroon">TPS</font> =  <a href="pcomps_1.html#K4" target="_self">TopSpaceMetr</a> <font color="Maroon">PM</font>;<br/>
<a NAME="E1:42"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" target="_self">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span>  st not <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> holds <br/> ex <font color="Olive">W</font>, <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" target="_self">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" target="_self">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" target="_self">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> ;<br/>
<a NAME="E3:42_1"/><i><font color="Green">E27</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T2" target="_self">Th2</a>, <a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>
<b>hence </b><a NAME="E2:42_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E4:42_1"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E5:42_1"/>
 ex <font color="Olive">W</font>, <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" target="_self">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">W</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon">V</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>;<br/>
<a NAME="E1:42_1_2"/><i><font color="Green">E32</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3" target="_self">Family_open_set</a> <font color="Maroon">PM</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E2:42_1_2"/><i><font color="Green">E33</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3" target="_self">Family_open_set</a> <font color="Maroon">PM</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E3:42_1_2"/><i><font color="Green">E48</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/>
<a NAME="E4:42_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>, <font color="Maroon">V</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" target="_self">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span> ;<br/>
<a NAME="E6:42_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_2_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:42_1_2_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">V</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> ;<br/>
<a NAME="E3:42_1_2_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:42_1_2_1_1"/><b>then </b>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E5:42_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E6:42_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:42_1_2_1"/>
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<b>take </b>
<font color="Maroon">V</font>
;<br/>

<b>thus </b><a NAME="E7:42_1_2"/>
( <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E2:42_1_2"><i><font color="Green">E33</font></i></a>, , , , , , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:42_1"/>
 ex <font color="Olive">W</font>, <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" target="_self">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span> st <br/>( <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">V</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:42"/>
 <a href="pcomps_1.html#K4" target="_self">TopSpaceMetr</a> <font color="Maroon">PM</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 <b>by </b><i><a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a></i>;<br/>


</div>
