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

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Olive" title="b1">d0</font></span>)</span>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">d0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/> <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Olive" title="b1">d0</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Olive" title="b2">A</font></span><br/><b>let </b><font color="Maroon" title="c2">d0</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">d0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/> <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Olive" title="b1">A</font></span><br/>



<b>set </b><font color="Maroon" title="c3">G</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c4">T</font> = <span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> ;<br/>
<a NAME="E1:48"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>;<br/>
<a NAME="E2:48"/><b>then </b><i><font color="Green" title="E20">A1</font></i>: 
 <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">d0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies  <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c5">A</font> )</span><br/>

<b>assume </b><a NAME="E3:48"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">d0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c5">A</font></span><br/>

<a NAME="E4:48"/><i><font color="Green" title="E22">A3</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">H</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">F</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> where <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">F</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span> : ( <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">P</font> where <font color="Olive" title="b3">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span>  &amp; <font color="Olive" title="b2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">P</font> where <font color="Olive" title="b3">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span>  ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tmap_1.html#D7" title="TMAP_1:def.7">TMAP_1:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">W</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span>  implies <font color="Maroon" title="c6">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:48_1_1"/>
<font color="Maroon" title="c6">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span></span><br/><b>then consider </b><font color="Maroon" title="c7">P</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E3:48_1_1"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P</font>
 <b>and </b><br/><a NAME="E4:48_1_1"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font>
 <b>and </b><br/><a NAME="E5:48_1_1"/>
<font color="Maroon" title="c7">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c1">D</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> <b>by </b><i><a class="txt" href="tex_1.html#E8"><i><font color="Green" title="E7">Lm1</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c9">H</font> = <font color="Maroon" title="c7">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">d0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><b>reconsider </b><font color="Maroon" title="c10">H</font> = <font color="Maroon" title="c7">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">d0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> ;<br/><a NAME="E8:48_1_1"/>
for <font color="Olive" title="b1">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font>  holds <br/>(  not <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> or  not <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> or  not <font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> )
 ;<br/><a NAME="E9:48_1_1"/><b>then </b>
not <font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> 
 ;<br/><a NAME="E10:48_1_1"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E11:48_1_1"/>
<font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">d0</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E12:48_1_1"/><b>then </b>
for <font color="Olive" title="b1">K</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font>  holds <br/>(  not <font color="Olive" title="b1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H</font> or  not <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> or  not <font color="Olive" title="b1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E13:48_1_1"/><b>then </b>
not <font color="Maroon" title="c10">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> 
 ;<br/><a NAME="E14:48_1_1"/><b>then </b><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c10">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E15:48_1_1"/>
( <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )
 <b>by </b><i><a class="txt" href="tex_1.html#E3:48"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="tex_1.html#E4:48_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E16:48_1_1"/><b>then </b>
( <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font> )
 <b>by </b><i><a class="txt" href="tex_1.html#E3:48"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="tex_1.html#E3:48_1_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>;<br/><b>hence </b><a NAME="E17:48_1_1"/>
<font color="Maroon" title="c6">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span>
 <b>by </b><i><a class="txt" href="tex_1.html#E1:48_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="tex_1.html#E10:48_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="tex_1.html#E14:48_1_1"><i><font color="Green" title="E27">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:48_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:48"/>
<span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T99" title="TMAP_1:th.99">TMAP_1:99</a></i>;<br/>
<a NAME="E6:48"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">D</font> : ( <font color="Maroon" title="c2">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="tex_1.html#E4:48"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E7:48"/><b>then </b><i><font color="Green" title="E23">A9</font></i>: 
 <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span>
 
<b>by </b><i><a class="txt" href="tex_1.html#E2:48"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c5">A</font></span>)</span> = 
<font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></i>
<br/>.= 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">D</font></span>)</span>
<b>by </b><i><a class="txt" href="tex_1.html#E7:48"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E9:48"/>
 <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tex_1.html#K3" title="TEX_1:func.3">STS</a> <font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c2">d0</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
