<?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>assume </b><a NAME="E1:39"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR4" title="COMPTS_1:NR.4">is_T4</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:39"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E3:39"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> or <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c4">V</font> = <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#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">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/><a NAME="E3:39_1_1"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c2">A</font> <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#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c4">V</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22" title="PRE_TOPC:th.22">PRE_TOPC:22</a></i>;<br/><a NAME="E4:39_1_1"/><b>then </b><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c4">V</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="urysohn1.html#E2:39"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/><a NAME="E5:39_1_1"/><i><font color="Green" title="E29">A7</font></i>: 
<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#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:39_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="urysohn1.html#E3:39_1_1"><i><font color="Green" title="E27">A5</font></i></a></i>;<br/><a NAME="E6:39_1_1"/>
<font color="Maroon" title="c3">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_1_1"/>
<font color="Maroon" title="c3">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon" title="c5">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_1_1_1"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:39_1_1_1"/>
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font> )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E3:39_1_1_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:39_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn1.html#E3:39"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c5">C</font>, <font color="Maroon" title="c6">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E8:39_1_1"/><i><font color="Green" title="E30">A9</font></i>: 
( <font color="Maroon" title="c5">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> &amp; <font color="Maroon" title="c4">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Q</font> &amp; <font color="Maroon" title="c5">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Q</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:39"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="urysohn1.html#E3:39"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="urysohn1.html#E4:39_1_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="urysohn1.html#E5:39_1_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="compts_1.html#D6" title="COMPTS_1:def.6">COMPTS_1:def 6</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">C</font>
;<br/><b>thus </b><a NAME="E9:39_1_1"/>
( <font color="Maroon" title="c5">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E8:39_1_1"><i><font color="Green" title="E30">A9</font></i></a></i>;<br/><a NAME="E10:39_1_1"/><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c6">Q</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="txt" href="urysohn1.html#E8:39_1_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="tops_1.html#T30" title="TOPS_1:th.30">TOPS_1:30</a></i>;<br/><a NAME="E11:39_1_1"/>
<font color="Maroon" title="c5">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="urysohn1.html#E8:39_1_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/><a NAME="E12:39_1_1"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Q</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="urysohn1.html#E10:39_1_1"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="tops_1.html#T31" title="TOPS_1:th.31">TOPS_1:31</a></i>;<br/><a NAME="E13:39_1_1"/><b>then </b>
( <font color="Maroon" title="c4">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Q</font> &amp; <font color="Maroon" title="c6">Q</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">C</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E8:39_1_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/><a NAME="E14:39_1_1"/><b>then </b><i><font color="Green" title="E32">A11</font></i>: 
<font color="Maroon" title="c4">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">C</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/><a NAME="E15:39_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 ;<br/><b>hence </b><a NAME="E16:39_1_1"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c5">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E14:39_1_1"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 ;<br/><div class="add"><b>take </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
;<br/><b>thus </b><a NAME="E2:39_1_2"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:39_1_2"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
