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

<span class="kw">set </span><font color="Maroon" title="c1">C</font> =  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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/>
<span class="kw">set </span><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="setfam_1.html#K9" title="SETFAM_1:func.9">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="E1:60_1_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E2:60_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">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="c4">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<a NAME="E4:60_1_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">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="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font>;<br/>
<a NAME="E6:60_1_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">A1</font> = <font color="Maroon" title="c5">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">A2</font> = <font color="Maroon" title="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">A3</font> = <font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>
<a NAME="E9:60_1_1"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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="c3">B0</font>,<font color="Maroon" title="c4">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">H</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span> implies <font color="Maroon" title="c12">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="c7">A1</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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/><span class="kw">assume </span><a NAME="E1:60_1_1_1"/>
<font color="Maroon" title="c12">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">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="c7">A1</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="E2:60_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <span class="p1">(<span class="default"><a href="tex_1.html#K2" title="TEX_1:func.2">ADTS</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></span>;<br/><a NAME="E3:60_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c5">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="setfam_1.html#K9" title="SETFAM_1:func.9">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#D7" title="TMAP_1:def.7">TMAP_1:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">P</font>, <font color="Maroon" title="c14">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="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw"> such that </span><br/><a NAME="E5:60_1_1_1"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:60_1_1_1"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c13">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="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">and </span><br/><a NAME="E7:60_1_1_1"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c14">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="60_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:60_1_1_1_1_1"/>
( ( <font color="Maroon" title="c13">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="c14">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="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c14">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="c13">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E7:60_1_1_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:60_1_1_1_1_1_1"/>
( <font color="Maroon" title="c13">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="c14">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">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"><span class="kw">hence </span><a NAME="E2:60_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:60_1_1_1_1_1_2"/>
( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c14">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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"><span class="kw">hence </span><a NAME="E2:60_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:60_1_1_1_1_1_3"/>
( <font color="Maroon" title="c13">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:60_1_1_1_1_1_3"/>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1_1_4"><span class="kw">case </span></a><a NAME="E1:60_1_1_1_1_1_4"/>
( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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"><span class="kw">hence </span><a NAME="E2:60_1_1_1_1_1_4"/>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:60_1_1_1"/>
<font color="Maroon" title="c12">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="c7">A1</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E12:60_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A8</font></span>: 
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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">H</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> implies <font color="Maroon" title="c12">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="c9">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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/><span class="kw">assume </span><a NAME="E1:60_1_1_2"/>
<font color="Maroon" title="c12">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">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="c9">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="E2:60_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></span>;<br/><a NAME="E3:60_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span> ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#D7" title="TMAP_1:def.7">TMAP_1:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">P</font>, <font color="Maroon" title="c14">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:60_1_1_2"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:60_1_1_2"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c13">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:60_1_1_2"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
<font color="Maroon" title="c14">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:60_1_1_2_1_1"/>
( ( <font color="Maroon" title="c13">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="c14">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="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> &amp; <font color="Maroon" title="c14">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="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c14">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="c13">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> ) or ( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> &amp; <font color="Maroon" title="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> ) or ( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> ) or ( <font color="Maroon" title="c13">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> &amp; <font color="Maroon" title="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:60_1_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E6:60_1_1_2"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="txt" href="#E7:60_1_1_2"><span class="lab"><font color="Green" title="E31">A11</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:60_1_1_2_1_1_1"/>
( <font color="Maroon" title="c13">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="c14">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">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"><span class="kw">hence </span><a NAME="E2:60_1_1_2_1_1_1"/>
<font color="Maroon" title="c12">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:60_1_1_2_1_1_2"/>
( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> &amp; <font color="Maroon" title="c14">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:60_1_1_2_1_1_2"/>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_3"><span class="kw">case </span></a><a NAME="E1:60_1_1_2_1_1_3"/>
( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c14">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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"><span class="kw">hence </span><a NAME="E2:60_1_1_2_1_1_3"/>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_4"><span class="kw">case </span></a><a NAME="E1:60_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
( <font color="Maroon" title="c13">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">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="E2:60_1_1_2_1_1_4"/>
<font color="Maroon" title="c7">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><span class="kw">hence </span><a NAME="E3:60_1_1_2_1_1_4"/>
<font color="Maroon" title="c12">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E1:60_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_5"><span class="kw">case </span></a><a NAME="E1:60_1_1_2_1_1_5"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> &amp; <font color="Maroon" title="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font></span><br/><div class="add"><a NAME="E2:60_1_1_2_1_1_5"/>
<font color="Maroon" title="c7">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><a NAME="E3:60_1_1_2_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c7">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:60_1_1_2_1_1_5"/>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E1:60_1_1_2_1_1_5"><span class="lab"><font color="Green" title="E32">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_6"><span class="kw">case </span></a><a NAME="E1:60_1_1_2_1_1_6"/><span class="lab"><font color="Green" title="E32">A14</font></span>: 
( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="E2:60_1_1_2_1_1_6"/>
<font color="Maroon" title="c7">A1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><a NAME="E3:60_1_1_2_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c7">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:60_1_1_2_1_1_6"/>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_1_1_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E1:60_1_1_2_1_1_6"><span class="lab"><font color="Green" title="E32">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_7"><span class="kw">case </span></a><a NAME="E1:60_1_1_2_1_1_7"/>
( <font color="Maroon" title="c13">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:60_1_1_2_1_1_7"/>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E9:60_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E5:60_1_1_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_8"><span class="kw">case </span></a><a NAME="E1:60_1_1_2_1_1_8"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A1</font> &amp; <font color="Maroon" title="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="E2:60_1_1_2_1_1_8"/>
<font color="Maroon" title="c7">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></span>;<br/><span class="kw">hence </span><a NAME="E3:60_1_1_2_1_1_8"/>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E5:60_1_1_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E1:60_1_1_2_1_1_8"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_2_1_1_9"><span class="kw">case </span></a><a NAME="E1:60_1_1_2_1_1_9"/>
( <font color="Maroon" title="c13">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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"><span class="kw">hence </span><a NAME="E2:60_1_1_2_1_1_9"/>
<font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E5:60_1_1_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:60_1_1_2"/>
<font color="Maroon" title="c12">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="c9">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:60_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A16</font></span>: 
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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="c9">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E15:60_1_1"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c9">A2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_1.html#T1" target="_self" title="TEX_1:th.1">Th1</a>, <a class="ref" href="tmap_1.html#T105" title="TMAP_1:th.105">TMAP_1:105</a></span>;<br/>
<a NAME="E16:60_1_1"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c9">A2</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_1.html#T3" target="_self" title="TEX_1:th.3">Th3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">H</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">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="c9">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c12">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:60_1_1_3"/>
<font color="Maroon" title="c12">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="c9">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span></span><br/><a NAME="E2:60_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c12">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="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A2</font> or <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or <font color="Maroon" title="c12">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:60_1_1_3"/>
<font color="Maroon" title="c12">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E9:60_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E16:60_1_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E15:60_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E18:60_1_1"/><span class="kw">then </span>
<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="c9">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E19:60_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A19</font></span>: 
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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="c9">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E14:60_1_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E33">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><div><span class="lab"><font color="Green" title="E34">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:60_1_1_4_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="setfam_1.html#K9" title="SETFAM_1:func.9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:60_1_1_4_1"/><span class="kw">then </span>
( <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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="tmap_1.html#T104" title="TMAP_1:th.104">TMAP_1:104</a></span>;<br/><span class="kw">hence </span><a NAME="E3:60_1_1_4_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E35">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_4_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:60_1_1_4_2"/>
<font color="Maroon" title="c9">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:60_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><a NAME="E2:60_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A23</font></span>: 
<font color="Maroon" title="c9">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><span class="kw">assume </span><a NAME="E3:60_1_1_4_2"/>
<font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">hence </span><a NAME="E4:60_1_1_4_2"/>
<font color="Maroon" title="c12">G</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_4_2"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E36">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_4_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:60_1_1_4_3"/>
<font color="Maroon" title="c9">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:60_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><a NAME="E2:60_1_1_4_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><span class="kw">assume </span><a NAME="E3:60_1_1_4_3"/>
<font color="Maroon" title="c12">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">hence </span><a NAME="E4:60_1_1_4_3"/>
<font color="Maroon" title="c12">G</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_4_3"><span class="lab"><font color="Green" title="E37">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E4:60_1_1_4"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E5:60_1_1_4"/><span class="kw">then </span>
<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="c9">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E19:60_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:60_1_1_4"/>
<font color="Maroon" title="c12">G</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1_4"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="txt" href="#E2:60_1_1_4"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E3:60_1_1_4"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E21:60_1_1"/><span class="lab"><font color="Green" title="E34">A26</font></span>: 
 <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="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E35">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_5"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E36">A28</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c12">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E9:60_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#D1" title="ZFMISC_1:def.1">ZFMISC_1:def 1</a></span>;<br/><a NAME="E2:60_1_1_5_1"/><span class="lab"><font color="Green" title="E37">A29</font></span>: 
not <font color="Maroon" title="c12">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:60_1_1_5_1"/><span class="lab"><font color="Green" title="E38">A30</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></span>;<br/><span class="kw">assume </span><a NAME="E4:60_1_1_5_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:60_1_1_5_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c11">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_5_1"><span class="lab"><font color="Green" title="E37">A29</font></span></a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></span>;<br/><a NAME="E6:60_1_1_5_1"/><span class="kw">then </span>
( <font color="Maroon" title="c12">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B0</font> or <font color="Maroon" title="c12">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1_5_1"><span class="lab"><font color="Green" title="E38">A30</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E7:60_1_1_5_1"/><span class="kw">then </span>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c4">B1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:60_1_1_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">take </span><font color="Maroon" title="c12">B</font> = <font color="Maroon" title="c11">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><div><span class="lab"><font color="Green" title="E37">A31</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_5_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:60_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="c9">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:60_1_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A32</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:60_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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E4:60_1_1_5_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_5_2"><span class="lab"><font color="Green" title="E38">A32</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E38">A33</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_5_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:60_1_1_5_3"/>
 <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="setfam_1.html#K9" title="SETFAM_1:func.9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:60_1_1_5_3"/><span class="kw">then </span>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:60_1_1_5_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:60_1_1"><span class="lab"><font color="Green" title="E34">A26</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:60_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="c9">A2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">A2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E19:60_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><a NAME="E5:60_1_1_5"/><span class="kw">then </span>
(  <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="c9">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="c9">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="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:60_1_1_5"/>
<font color="Maroon" title="c12">B</font> is  <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1_5"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E2:60_1_1_5"><span class="lab"><font color="Green" title="E37">A31</font></span></a>, <a class="txt" href="#E1:60_1_1_5"><span class="lab"><font color="Green" title="E36">A28</font></span></a>, <a class="ref" href="tops_1.html#T84" title="TOPS_1:th.84">TOPS_1:84</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">take </span>
<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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c12">A2</font> = <font color="Maroon" title="c9">A2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="60_1_1_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:60_1_1_6_1"/>
<font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:60_1_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">B0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:60_1_1_6_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:60_1_1_6"/>
( <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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></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>, <a class="ref" href="tmap_1.html#T105" title="TMAP_1:th.105">TMAP_1:105</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E24:60_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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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="setfam_1.html#K9" title="SETFAM_1:func.9">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="c5">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="c7">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E22:60_1_1"><span class="lab"><font color="Green" title="E35">A27</font></span></a>, <a class="txt" href="#E20:60_1_1"><span class="lab"><font color="Green" title="E33">A20</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
