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

<b>let </b><font color="Maroon" title="c1">S</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a><br/> for <font color="Olive" title="b2">K</font> being   <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Olive" title="b1">T</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">T</font> holds <br/> ( <font color="Olive" title="b3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> iff for <font color="Olive" title="b4">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">T</font>  st <font color="Olive" title="b4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> holds <br/><font color="Olive" title="b3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>let </b><font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being   <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> holds <br/> ( <font color="Olive" title="b2">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> iff for <font color="Olive" title="b3">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Olive" title="b2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>let </b><font color="Maroon" title="c3">BB</font> be    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font> holds <br/> ( <font color="Olive" title="b1">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> iff for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> holds <br/><font color="Olive" title="b1">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> iff for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>







<a NAME="E1:53"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c3">BB</font> <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 <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D5" title="CANTOR_1:def.5">CANTOR_1:def 5</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) implies <font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><b>let </b><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> implies <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>assume </b><a NAME="E2:53_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><a NAME="E3:53_1"/><b>then </b>
<font color="Maroon" title="c5">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:53"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E4:53_1"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T30" title="TOPS_1:th.30">TOPS_1:30</a></i>;<br/><b>hence </b><a NAME="E5:53_1"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E1:53_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:53"/><i><font color="Green" title="E28">A3</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">BB</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">C</font> =  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c3">BB</font> as    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c2">T</font> <b>by </b><i><a class="ref" href="yellow_9.html#T23" target="_self" title="YELLOW_9:th.23">Th23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> implies <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/><b>assume </b><a NAME="E1:53_2"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><b>then consider </b><font color="Maroon" title="c7">Y</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">T</font><b> such that </b><br/><a NAME="E3:53_2"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">BB</font> &amp; <font color="Maroon" title="c7">Y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c7">Y</font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D4" title="CANTOR_1:def.4">CANTOR_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c7">Y</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">T</font> ;<br/><b>reconsider </b><font color="Maroon" title="c9">CY</font> =  <a href="setfam_1.html#K7" title="SETFAM_1:func.7">COMPLEMENT</a> <font color="Maroon" title="c8">Y</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">T</font> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> = $1 <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/><i><font color="Green" title="E30">A5</font></i>: <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> = 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">CY</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_9.html#E3:53_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="yellow_8.html#T16" title="YELLOW_8:th.16">YELLOW_8:16</a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] </span>}</span> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self" title="YELLOW_9:th.3">Th3</a></i>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>)</span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] </span>}</span> 
<b>from </b><i><a class="ref" href="yellow_9.html#S4" target="_self" title="YELLOW_9:sch.4">YELLOW_9:sch 4</a>()</i>
;<br/><b>set </b><font color="Maroon" title="c10">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> </span>}</span> ;<br/><a NAME="E7:53_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">x</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="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> </span>}</span>  or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> )</span><br/>

<b>assume </b><a NAME="E1:53_2_2"/>
<font color="Maroon" title="c11">x</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="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><br/>

<a NAME="E2:53_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> )
 
;<br/>
<b>hence </b><a NAME="E3:53_2_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c11">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">S</font> ;<br/><b>reconsider </b><font color="Maroon" title="c12">X</font> = <font color="Maroon" title="c11">X</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">S</font> ;<br/><a NAME="E10:53_2"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c12">X</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">a</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> or <font color="Maroon" title="c13">a</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span><br/>

<b>assume </b><a NAME="E1:53_2_3"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">a</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<a NAME="E2:53_2_3"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> st <br/>( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> )
 
;<br/>
<b>hence </b><a NAME="E3:53_2_3"/>
<font color="Maroon" title="c13">a</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:53"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="yellow_9.html#E3:53_2"><i><font color="Green" title="E29">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E11:53_2"/><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c8">Y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E3:53_2"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> = <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default">$1 <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>;<br/><a NAME="E12:53_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">y</font>) where <font color="Olive" title="b1">y</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21" title="FRAENKEL:sch.21">FRAENKEL:sch 21</a>(<a class="txt" href="yellow_9.html#E11:53_2"><i><font color="Green" title="E32">A7</font></i></a>);<br/></i><b>hence </b><a NAME="E13:53_2"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="yellow_9.html#E6:53_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="yellow_9.html#E10:53_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="tops_2.html#T28" title="TOPS_2:th.28">TOPS_2:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:53"/>
<font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="yellow_9.html#T33" target="_self" title="YELLOW_9:th.33">Th33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
