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<span class="kw">let </span><font color="Maroon" title="c1">X</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="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> 1</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">T</font> =  <a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">CT</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <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"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></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"> for <font color="Olive" title="b1">U</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span>  st <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> holds <br/> ex <font color="Olive" title="b2">W</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">U</font> be   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U</font> implies  ex <font color="Olive" title="b1">W</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )</span><br/><span class="kw">assume </span><a NAME="E1:17_1"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">take </span><font color="Maroon" title="c6">W</font> = <font color="Maroon" title="c5">U</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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font> &amp; <font color="Maroon" title="c6">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><a NAME="E2:17_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
 <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="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></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"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">W9</font> = <font color="Maroon" title="c5">U</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/><a NAME="E4:17_1"/>
<font color="Maroon" title="c7">W9</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/><a NAME="E5:17_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">W9</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c7">W9</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="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a>, <a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></span>;<br/><a NAME="E6:17_1"/><span class="kw">then </span>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c7">W9</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"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_1.html#T16" title="TOPGEN_1:th.16">TOPGEN_1:16</a></span>;<br/><a NAME="E7:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c7">W9</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E8:17_1"/>
<font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">W</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><a NAME="E9:17_1"/><span class="kw">then </span>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c6">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="topdim_1.html#T1" target="_self" title="TOPDIM_1:th.1">Th1</a></span>;<br/><a NAME="E10:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c6">W</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"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ;<br/><a NAME="E11:17_1"/>
<span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p3">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#D1" target="_self" title="TOPDIM_1:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:17_1"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font> &amp; <font color="Maroon" title="c6">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E10:17_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <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>
<a NAME="E2:17"/><span class="kw">then </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#D1" target="_self" title="TOPDIM_1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:17"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topdim_1.html#K1" title="TOPDIM_1:func.1">Seq_of_ind</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1" title="COMPTS_1:func.1">1TopSp</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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