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

<b>reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:37_1_1"/>
(  not <font color="Maroon">Y0</font> is <a href="t_0topsp.html#V2">T_0</a> or <font color="Maroon">Y0</font> is <a href="struct_0.html#V3">empty</a> or for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>  st <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y0</font> &amp; <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y0</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; ( for <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> or  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> or <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> ) ) holds <br/> ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">Y0</font> is <a href="t_0topsp.html#V2">T_0</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_1_1_1_1"/><i><font color="Green">E18</font></i>: 
not <font color="Maroon">Y0</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>;<br/><b>assume </b><a NAME="E2:37_1_1_1_1"/>
( <font color="Maroon">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y0</font> &amp; <font color="Maroon">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y0</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">x0</font> = <font color="Maroon">x</font>, <font color="Maroon">y0</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y0</font> ;<br/><a NAME="E4:37_1_1_1_1"/><i><font color="Green">E19</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>assume </b><a NAME="E5:37_1_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1_1_1_1"/>
(  ex <font color="Olive">E0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">E0</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">E0</font> &amp; not <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">E0</font> ) or  ex <font color="Olive">F0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">F0</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">F0</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">F0</font> ) )
 </span><b>by </b><i>, , , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1_1"/>
 ex <font color="Olive">E0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">E0</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">E0</font> &amp; not <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">E0</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">E0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font><b> such that </b><br/><a NAME="E3:37_1_1_1_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">E0</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:37_1_1_1_1_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 <b>and </b><br/><a NAME="E5:37_1_1_1_1_1_1_1"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">E</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:37_1_1_1_1_1_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">E</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E3:37_1_1_1_1_1_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">E0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">E</font> = <font color="Maroon">E</font>;<br/><b>thus </b><a NAME="E4:37_1_1_1_1_1_1_1_1"/>
<font color="Maroon">E</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tsp_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E5:37_1_1_1_1_1_1_1_1"/>
<font color="Maroon">E</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E6:37_1_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="tsp_1.html#T2" target="_self">Th2</a>, </i>;<br/><b>thus </b><a NAME="E7:37_1_1_1_1_1_1_1_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:37_1_1_1_1_1_1_1_1"><i><font color="Green">E26</font></i></a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:37_1_1_1_1_1_1_1"/>
(  ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> ) or  ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_1_1_1_2"/>
 ex <font color="Olive">F0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">F0</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">F0</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">F0</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">F0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font><b> such that </b><br/><a NAME="E3:37_1_1_1_1_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">F0</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:37_1_1_1_1_1_1_2"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F0</font>
 <b>and </b><br/><a NAME="E5:37_1_1_1_1_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F0</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:37_1_1_1_1_1_1_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E3:37_1_1_1_1_1_1_2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E4:37_1_1_1_1_1_1_2"><i><font color="Green">E28</font></i></a>, </i>;<br/><b>take </b><font color="Maroon">F</font> = <font color="Maroon">F</font>;<br/><b>thus </b><a NAME="E4:37_1_1_1_1_1_1_2_1"/>
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tsp_1.html#D2" target="_self">Def2</a></i>;<br/><b>thus </b><a NAME="E5:37_1_1_1_1_1_1_2_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="tsp_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:37_1_1_1_1_1_1_2_1"/>
<font color="Maroon">F</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E7:37_1_1_1_1_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:37_1_1_1_1_1_1_2"/>
(  ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> ) or  ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:37_1_1_1_1"/>
(  ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> ) or  ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) )
 ;<br/></div>
<b>end;</b></div>


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

<b>assume </b><a NAME="E3:37_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">Y0</font> is <a href="struct_0.html#V3">empty</a> or for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>  st <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y0</font> &amp; <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y0</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; ( for <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> or  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> or <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> ) ) holds <br/> ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_1_2"/><i><font color="Green">E38</font></i>: 
not <font color="Maroon">Y0</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y0</font>, <font color="Maroon">y0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y0</font>;<br/><a NAME="E2:37_1_1_2"/><i><font color="Green">E39</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E3:37_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E4:37_1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> )
 ;<br/><a NAME="E5:37_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x0</font>, <font color="Maroon">y</font> = <font color="Maroon">y0</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/><b>assume </b><a NAME="E7:37_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y0</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_2_1_1"/>
(  ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> ) or  ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_2_1_1_1"/>
 ex <font color="Olive">E</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">E</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">E</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:37_1_1_2_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">E</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:37_1_1_2_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>and </b><br/><a NAME="E5:37_1_1_2_1_1_1"/><i><font color="Green">E61</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">E0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font><b> such that </b><br/><a NAME="E2:37_1_1_2_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">E0</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E3:37_1_1_2_1_1_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">E0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tsp_1.html#T2" target="_self">Th2</a>, </i>;<br/><b>take </b><font color="Maroon">E0</font> = <font color="Maroon">E0</font>;<br/><b>thus </b><a NAME="E4:37_1_1_2_1_1_1_1"/>
<font color="Maroon">E0</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:37_1_1_2_1_1_1_1"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_1_2_1_1_1_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 ;<br/><a NAME="E2:37_1_1_2_1_1_1_1_1"/>
<font color="Maroon">E</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E3:37_1_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tsp_1.html#D2" target="_self">Def2</a>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:37_1_1_2_1_1_1_1"/>
not <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:37_1_1_2_1_1_1"/>
(  ex <font color="Olive">E0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">E0</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">E0</font> &amp; not <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">E0</font> ) or  ex <font color="Olive">F0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">F0</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">F0</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">F0</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_2_1_1_2"/>
 ex <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:37_1_1_2_1_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:37_1_1_2_1_1_2"/><i><font color="Green">E93</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:37_1_1_2_1_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">F0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font><b> such that </b><br/><a NAME="E2:37_1_1_2_1_1_2_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">F0</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E3:37_1_1_2_1_1_2_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">F0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tsp_1.html#T2" target="_self">Th2</a>, </i>;<br/><b>take </b><font color="Maroon">F0</font> = <font color="Maroon">F0</font>;<br/><b>thus </b><a NAME="E4:37_1_1_2_1_1_2_1"/>
<font color="Maroon">F0</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1_1_2_1_1_2_1_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F0</font>
 ;<br/><a NAME="E2:37_1_1_2_1_1_2_1_1"/>
<font color="Maroon">F</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E3:37_1_1_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:37_1_1_2_1_1_2_1"/>
not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F0</font>
 ;<br/><b>thus </b><a NAME="E7:37_1_1_2_1_1_2_1"/>
<font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F0</font>
 <b>by </b><i>, <a class="ref" href="tsp_1.html#D2" target="_self">Def2</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:37_1_1_2_1_1_2"/>
(  ex <font color="Olive">E0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">E0</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">E0</font> &amp; not <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">E0</font> ) or  ex <font color="Olive">F0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">F0</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">F0</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">F0</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:37_1_1_2"/>
(  ex <font color="Olive">E0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">E0</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">E0</font> &amp; not <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">E0</font> ) or  ex <font color="Olive">F0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> st <br/>( <font color="Olive">F0</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Olive">F0</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Olive">F0</font> ) )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:37_1_1"/>
<font color="Maroon">Y0</font> is <a href="t_0topsp.html#V2">T_0</a>
 <b>by </b><i/>;<br/>


</div>
