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

<b>set </b><font color="Maroon" title="c1">C</font> =  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c2">Y</font> =  <a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>;<br/>
<a NAME="E1:56_1_1"/><i><font color="Green" title="E24">A1</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E2:56_1_1"/><b>then </b><i><font color="Green" title="E25">A2</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c3">A</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <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#K2" title="TEX_1:func.2">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E1:56_1_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">Y1</font> = <span class="p1">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font>;<br/>
<a NAME="E4:56_1_1"/><i><font color="Green" title="E26">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></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/>
<b>reconsider </b><font color="Maroon" title="c5">A1</font> = <font color="Maroon" title="c3">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</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/>
<a NAME="E6:56_1_1"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c5">A1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T105" title="TMAP_1:th.105">TMAP_1:105</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">H</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">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> implies <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c5">A1</font>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:56_1_1_1"/>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</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">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c5">A1</font>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span></span><br/><a NAME="E2:56_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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#K2" title="TEX_1:func.2">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></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/><a NAME="E3:56_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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#K2" title="TEX_1:func.2">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></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/><b>then consider </b><font color="Maroon" title="c7">P</font>, <font color="Maroon" title="c8">Q</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#K2" title="TEX_1:func.2">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:56_1_1_1"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>and </b><br/><a NAME="E6:56_1_1_1"/><i><font color="Green" title="E29">A6</font></i>: 
( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_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="E1:56_1_1_1_1_1"/>
( ( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c8">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) or ( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c8">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="#E6:56_1_1_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:56_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E5:56_1_1_1"><i><font color="Green" title="E28">A5</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:56_1_1_1_1_1_2"/>
( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c8">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E5:56_1_1_1"><i><font color="Green" title="E28">A5</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:56_1_1_1_1_1_3"/>
( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></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">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font></span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_1_1_1_3"/>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="#E5:56_1_1_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:56_1_1_1_1_1_4"/>
( <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c8">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></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">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_1_1_1_4"/>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E5:56_1_1_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:56_1_1_1"/>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c5">A1</font>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</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="E8:56_1_1"/><b>then </b><i><font color="Green" title="E28">A7</font></i>: 
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#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c5">A1</font>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">Y2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )</span><br/>

<a NAME="E9:56_1_1"/><i><font color="Green" title="E29">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</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/>
<b>reconsider </b><font color="Maroon" title="c7">A2</font> = <font color="Maroon" title="c5">A1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></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/>
<b>set </b><font color="Maroon" title="c8">A3</font> = <font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>
<a NAME="E11:56_1_1"/><i><font color="Green" title="E30">A9</font></i>: 
( <font color="Maroon" title="c7">A2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c7">A2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="#E6:56_1_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="tex_1.html#T1" target="_self" title="TEX_1:th.1">Th1</a>, <a class="ref" href="tex_1.html#T3" target="_self" title="TEX_1:th.3">Th3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">H</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="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> implies <font color="Maroon" title="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c7">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> )</span><br/><b>assume </b><a NAME="E1:56_1_1_2"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></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="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c7">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span></span><br/><a NAME="E2:56_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</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/><a NAME="E3:56_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span></span>)</span> where <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> : ( <font color="Olive" title="b1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Olive" title="b2">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</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/><b>then consider </b><font color="Maroon" title="c10">P</font>, <font color="Maroon" title="c11">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span><b> such that </b><br/><a NAME="E5:56_1_1_2"/><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:56_1_1_2"/><i><font color="Green" title="E32">A12</font></i>: 
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_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="E1:56_1_1_2_1_1"/>
( ( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or ( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> ) or ( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> ) or ( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> ) or ( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) or ( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) or ( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="#E8:56_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="#E6:56_1_1_2"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:56_1_1_2_1_1_1"/>
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E5:56_1_1_2"><i><font color="Green" title="E31">A11</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:56_1_1_2_1_1_2"/>
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font></span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font>
 <b>by </b><i><a class="txt" href="#E5:56_1_1_2"><i><font color="Green" title="E31">A11</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:56_1_1_2_1_1_3"/>
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_2_1_1_3"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E5:56_1_1_2"><i><font color="Green" title="E31">A11</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_1_1_4"><b>case </b></a><a NAME="E1:56_1_1_2_1_1_4"/><i><font color="Green" title="E33">A13</font></i>: 
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:56_1_1_2_1_1_4"/>
<font color="Maroon" title="c5">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1_2_1_1_4"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E5:56_1_1_2"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="#E1:56_1_1_2_1_1_4"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_1_1_5"><b>case </b></a><a NAME="E1:56_1_1_2_1_1_5"/><i><font color="Green" title="E33">A14</font></i>: 
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font></span><br/><div class="add"><a NAME="E2:56_1_1_2_1_1_5"/>
<font color="Maroon" title="c5">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><a NAME="E3:56_1_1_2_1_1_5"/><b>then </b>
<font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:56_1_1_2_1_1_5"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font>
 <b>by </b><i><a class="txt" href="#E5:56_1_1_2"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="#E1:56_1_1_2_1_1_5"><i><font color="Green" title="E33">A14</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_1_1_6"><b>case </b></a><a NAME="E1:56_1_1_2_1_1_6"/><i><font color="Green" title="E33">A15</font></i>: 
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:56_1_1_2_1_1_6"/>
<font color="Maroon" title="c5">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><a NAME="E3:56_1_1_2_1_1_6"/><b>then </b>
<font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E4:56_1_1_2_1_1_6"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E5:56_1_1_2"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="#E1:56_1_1_2_1_1_6"><i><font color="Green" title="E33">A15</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_1_1_7"><b>case </b></a><a NAME="E1:56_1_1_2_1_1_7"/>
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_2_1_1_7"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E4:56_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="#E9:56_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="#E5:56_1_1_2"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_1_1_8"><b>case </b></a><a NAME="E1:56_1_1_2_1_1_8"/>
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:56_1_1_2_1_1_8"/><b>then </b>
( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:56_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="#E5:56_1_1_2"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1_2_1_1_8"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_2_1_1_9"><b>case </b></a><a NAME="E1:56_1_1_2_1_1_9"/>
( <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c11">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><b>hence </b><a NAME="E2:56_1_1_2_1_1_9"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E4:56_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="#E5:56_1_1_2"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</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></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:56_1_1_2"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c7">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</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="E13:56_1_1"/><b>then </b><i><font color="Green" title="E31">A16</font></i>: 
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"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c7">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">H</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="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c7">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> implies <font color="Maroon" title="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> )</span><br/><b>assume </b><a NAME="E1:56_1_1_3"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c7">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span></span><br/><a NAME="E2:56_1_1_3"/><b>then </b>
( <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A2</font> or <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1_3"/>
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:56_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="#E9:56_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="#E11:56_1_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="pre_topc.html#T5" title="PRE_TOPC:th.5">PRE_TOPC:5</a>, <a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</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="E15:56_1_1"/><b>then </b>
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c7">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E16:56_1_1"/><b>then </b><i><font color="Green" title="E32">A17</font></i>: 
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"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c7">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="#E13:56_1_1"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E17:56_1_1"/>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">B0</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>, <font color="Maroon" title="c10">B1</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">B</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">B0</font>,<font color="Maroon" title="c10">B1</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E9:56_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></i>;<br/>
<div><i><font color="Green" title="E33">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_4"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">A2</font> = <font color="Maroon" title="c7">A2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">A2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">A2</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="c12">A2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1_1_4_1"/>
<font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:56_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c9">B0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1_4_1"/>
contradiction
 ; <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="E2:56_1_1_4"/>
( <font color="Maroon" title="c12">A2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">A2</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="c12">A2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E6:56_1_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="#E9:56_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="tex_1.html#T1" target="_self" title="TEX_1:th.1">Th1</a>, <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><b>end;</b></div>
<div><i><font color="Green" title="E34">A19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_5"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">B</font> = <font color="Maroon" title="c11">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">B</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a></span><br/><a NAME="E1:56_1_1_5"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c7">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E16:56_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E2:56_1_1_5"/><b>then </b><i><font color="Green" title="E35">A20</font></i>: 
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A2</font> or  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><div><i><font color="Green" title="E36">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1_1_5_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:56_1_1_5_1"/><b>then </b>
 <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">B</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:56_1_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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><div><i><font color="Green" title="E37">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1_1_5_2"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:56_1_1_5_2"/><b>then </b>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">B</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1_5_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1_1_5_3"/><i><font color="Green" title="E38">A23</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>reconsider </b><font color="Maroon" title="c13">I</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E4:56_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="#E9:56_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#D1" title="ZFMISC_1:def.1">ZFMISC_1:def 1</a></i>;<br/><a NAME="E3:56_1_1_5_3"/>
not <font color="Maroon" title="c13">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:56_1_1_5_3"/><b>then </b>
( <font color="Maroon" title="c13">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">B</font> )
 <b>by </b><i><a class="txt" href="#E1:56_1_1_5_3"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a>, <a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/><a NAME="E5:56_1_1_5_3"/><b>then </b>
( <font color="Maroon" title="c13">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B0</font> or <font color="Maroon" title="c13">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B1</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E6:56_1_1_5_3"/><b>then </b>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B0</font> or  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B1</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:56_1_1_5_3"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E6:56_1_1_5"/>
<font color="Maroon" title="c12">B</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a>
 <b>by </b><i><a class="txt" href="#E2:56_1_1_5"><i><font color="Green" title="E35">A20</font></i></a>, <a class="txt" href="#E3:56_1_1_5"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="#E4:56_1_1_5"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="tops_1.html#T84" title="TOPS_1:th.84">TOPS_1:84</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></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="c12">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>assume </b><a NAME="E1:56_1_1_6"/>
<font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><a NAME="E2:56_1_1_6"/><b>then </b><i><font color="Green" title="E35">A24</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c7">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p3"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="#E16:56_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><div><i><font color="Green" title="E36">A25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1_1_6_1"/>
( <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><a NAME="E2:56_1_1_6_1"/><b>then </b>
( ( <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> or <font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> ) &amp;  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="#E9:56_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></i>;<br/><b>hence </b><a NAME="E3:56_1_1_6_1"/>
<font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <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><div><i><font color="Green" title="E37">A26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1_1_6_2"/><i><font color="Green" title="E38">A27</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><a NAME="E2:56_1_1_6_2"/>
<font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E16:56_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:56_1_1_6_2"/><b>then </b>
<font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E4:56_1_1_6_2"/>
<font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="#E1:56_1_1_6_2"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_6_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:56_1_1_6_3"/><i><font color="Green" title="E38">A28</font></i>: 
<font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><a NAME="E2:56_1_1_6_3"/>
<font color="Maroon" title="c7">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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"><span class="p3">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p4">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p5"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E16:56_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/><a NAME="E3:56_1_1_6_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">A2</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>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><b>hence </b><a NAME="E4:56_1_1_6_3"/>
<font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="#E1:56_1_1_6_3"><i><font color="Green" title="E38">A28</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="E6:56_1_1_6"/>
<font color="Maroon" title="c12">G</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="#E2:56_1_1_6"><i><font color="Green" title="E35">A24</font></i></a>, <a class="txt" href="#E3:56_1_1_6"><i><font color="Green" title="E36">A25</font></i></a>, <a class="txt" href="#E4:56_1_1_6"><i><font color="Green" title="E37">A26</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</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="E23:56_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="tdlat_3.html#V1" title="TDLAT_3:attr.1">discrete</a> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="tdlat_3.html#V2" title="TDLAT_3:attr.2">anti-discrete</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p3">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <span class="p4"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 <b>by </b><i><a class="txt" href="#E20:56_1_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="#E21:56_1_1"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="tex_1.html#D6" target="_self" title="TEX_1:def.6">Def6</a>, <a class="ref" href="tdlat_3.html#T20" title="TDLAT_3:th.20">TDLAT_3:20</a>, <a class="ref" href="tdlat_3.html#T23" title="TDLAT_3:th.23">TDLAT_3:23</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>
