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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:36"/>
<font color="Maroon">T</font> is <a href="pcomps_1.html#V3">metrizable</a>
 ;<br/>

<b>then consider </b><font color="Maroon">metr</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:36"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">metr</font> <a href="pcomps_1.html#R1">is_metric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp;  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,<font color="Maroon">metr</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="pcomps_1.html#D9">PCOMPS_1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">PM</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="metric_1.html#L1">MetrStruct</a> <b> such that </b><br/><a NAME="E5:36"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">PM</font> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K5">SpaceMetr</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,<font color="Maroon">metr</font>
 ;<br/>
<b>set </b><font color="Maroon">TPM</font> =  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font>;<br/>
<a NAME="E6:36"/><i><font color="Green">E58</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E7:36"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><br/> for <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">D</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> <a href="subset_1.html#K3">`</a>  holds <br/> ex <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">D</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">D</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K3">`</a>  )
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> <b>by </b><i>, , <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E3:36_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">D</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">D</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">D</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E4:36_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">D</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E5:36_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span> <b>by </b><i>, , <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E7:36_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">PM</font>
 
<b>by </b><i>, , , , <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E8:36_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">D</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E9:36_1"/><b>then </b><i><font color="Green">E76</font></i>: 
( <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">D</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">D</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E10:36_1"/>
 ex <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Olive">r1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">D</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Olive">r1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">D</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span>;<br/><b>assume </b><a NAME="E1:36_1_1_1"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><a NAME="E2:36_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">PM</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:36_1_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">r2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/><a NAME="E5:36_1_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:36_1_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>thus </b><a NAME="E8:36_1_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">D</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:36_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:36_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E12:36_1"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">D</font> )
 ;<br/>
<b>set </b><font color="Maroon">r2</font> = <font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2;<br/>
<b>set </b><font color="Maroon">r4</font> = <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2;<br/>
<a NAME="E13:36_1"/>
<font color="Maroon">r1</font> <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="E14:36_1"/><b>then </b>
( <font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</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="E15:36_1"/><b>then </b><i><font color="Green">E110</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</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"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp;  <a href="metric_1.html#K2">dist</a> <font color="Maroon">p</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T1">METRIC_1:1</a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E16:36_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1_2"/>
not <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> <b>by </b><i/>;<br/>
<a NAME="E5:36_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">PM</font>
 
;<br/>
<a NAME="E6:36_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E7:36_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:36_1_2"/><b>then </b>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="metric_1.html#T13">METRIC_1:13</a></i>;<br/>
<a NAME="E9:36_1_2"/><b>then </b>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:36_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E11:36_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:36_1"/><i><font color="Green">E113</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1_3"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:36_1_3"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">PM</font> <b>by </b><i><a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E5:36_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:36_1_3"/><b>then </b>
(  <a href="metric_1.html#K2">dist</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2 &amp;  <a href="metric_1.html#K2">dist</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a>, <a class="ref" href="metric_1.html#T13">METRIC_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:36_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>;<br/>
<a NAME="E18:36_1"/><i><font color="Green">E116</font></i>: 
( <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <font color="Maroon">PM</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</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">Family_open_set</a> <font color="Maroon">PM</font> )
 
<b>by </b><i>, , <a class="ref" href="pcomps_1.html#T33">PCOMPS_1:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>, <font color="Maroon">B</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">PM</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

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

<b>thus </b><a NAME="E20:36_1"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i>, , , , , <a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</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>
<a NAME="E8:36"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st not <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> holds <br/> ex <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; not <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> &amp; not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:36_2"/><i><font color="Green">E117</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font>, <font color="Maroon">q</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span> <b>by </b><i>, , <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E3:36_2"/>
(  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font> <a href="compts_1.html#NR2" target="_self">is_T2</a>  &amp; <font color="Maroon">p</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span> &amp; <font color="Maroon">q</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, , <a class="ref" href="pcomps_1.html#T38">PCOMPS_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span>, <font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">PM</font></span>)</span><b> such that </b><br/><a NAME="E5:36_2"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font>, <font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i>, , <a class="ref" href="pcomps_2.html#T8">PCOMPS_2:8</a></i>;<br/>
<a NAME="E7:36_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, , , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E8:36_2"/><b>then </b>
( <font color="Maroon">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E9:36_2"/>
 ex <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:36"/>
( <font color="Maroon">T</font> <a href="compts_1.html#NR3" target="_self">is_T3</a>  &amp; <font color="Maroon">T</font> <a href="urysohn1.html#NR1" target="_self">is_T1</a>  )
 <b>by </b><i>, <a class="ref" href="compts_1.html#D5">COMPTS_1:def 5</a>, <a class="ref" href="urysohn1.html#D9">URYSOHN1:def 9</a></i>;<br/>


</div>
