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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">d0</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>





<b>set </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">T</font> = <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font> : ( <font color="Maroon">d0</font> <a href="hidden.html#R2">in</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:45"/>
(  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font></span>)</span> #) implies ( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> holds   <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E16</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font></span>)</span> #)
 ;<br/>

<b>hence </b><a NAME="E2:45_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">P</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/><a NAME="E2:45_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:45_1_1"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="tex_1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E4:45_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><a NAME="E5:45_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E6:45_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E7:45_1_1"/>
( not  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="ref" href="pcomps_1.html#T2">PCOMPS_1:2</a></i>;<br/><a NAME="E8:45_1_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="tex_1.html#T4" target="_self">Th4</a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E9:45_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><b>hence </b><a NAME="E10:45_1_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="txt" href="tex_1.html#E6:45_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:45_1"/>
for <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> holds   <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>


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

<b>assume </b><a NAME="E2:45"/><i><font color="Green">E24</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 ;<br/>

<b>assume </b><a NAME="E3:45"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">A</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> holds   <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><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 <font color="Maroon">Y</font>;<br/><b>thus </b><a NAME="E1:45_2"/>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> implies <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_2_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:45_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E3:45_2_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:45_2_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:45_2"/>
( not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> implies <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_2_2"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>assume </b><a NAME="E2:45_2_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>assume </b><a NAME="E3:45_2_2"/><i><font color="Green">E51</font></i>: 
not <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E4:45_2_2"/>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d0</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="txt" href="tex_1.html#E2:45"><i><font color="Green">E24</font></i></a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>hence </b><a NAME="E5:45_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_1.html#E3:45"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:45"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" target="_self">STS</a> <font color="Maroon">D</font>,<font color="Maroon">d0</font></span>)</span> #)
 <b>by </b><i>, <a class="ref" href="tex_1.html#T4" target="_self">Th4</a></i>;<br/>


</div>
