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

<b>let </b><font color="Maroon" title="c1">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>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>set </b><font color="Maroon" title="c3">V</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c4">Y</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> ;<br/>
<a NAME="E1:103"/><i><font color="Green" title="E51">A1</font></i>: 
 <a href="pcomps_1.html#K1" title="PCOMPS_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">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c5">X1</font> =  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E1:103_1"/>
 <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E2:103_1"/>
 <a href="pcomps_1.html#K1" title="PCOMPS_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">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:103_2"/>
<font color="Maroon" title="c5">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c6">S</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:103_2"/><i><font color="Green" title="E52">A2</font></i>: 
( <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c6">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> )
 ;<br/><a NAME="E4:103_2"/>
<font color="Maroon" title="c5">Z</font> is   <a href="prob_1.html#NM1" title="PROB_1:NM.1">Field_Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E3:103_2"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="topgen_4.html#T13" target="_self" title="TOPGEN_4:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E5:103_2"/>
 <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="c5">Z</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:103"/><b>then </b><i><font color="Green" title="E52">A3</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="topgen_4.html#E1:103"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E4:103"/><i><font color="Green" title="E53">A4</font></i>: 
for <font color="Olive" title="b1">BSeq</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>  st ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">S</font> where <font color="Olive" title="b3">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">S</font> &amp; <font color="Olive" title="b3">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span>  ) holds <br/> <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Olive" title="b1">BSeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">S</font> where <font color="Olive" title="b2">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b2">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">BSeq</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:103_3"/><i><font color="Green" title="E54">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c5">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">S</font> where <font color="Olive" title="b2">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b2">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:103_3_1"/><i><font color="Green" title="E55">A6</font></i>: 
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c7">S</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:103_3_1"/><i><font color="Green" title="E56">A7</font></i>: 
( <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Maroon" title="c7">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> )
 ;<br/><a NAME="E4:103_3_1"/><i><font color="Green" title="E57">A8</font></i>: 
( <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Maroon" title="c7">S</font> is   <a href="topgen_4.html#NM1" title="TOPGEN_4:NM.1">SigmaField</a> of <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="txt" href="topgen_4.html#E3:103_3_1"><i><font color="Green" title="E56">A7</font></i></a>, <a class="ref" href="topgen_4.html#T13" target="_self" title="TOPGEN_4:th.13">Th13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:103_3_1_1"/>
<font color="Maroon" title="c5">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:103_3"><i><font color="Green" title="E54">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E2:103_3_1_1"/>
<font color="Maroon" title="c5">BSeq</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:103_3_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:103_3_1"/>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c5">BSeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E4:103_3_1"><i><font color="Green" title="E57">A8</font></i></a>, <a class="ref" href="prob_1.html#D8" title="PROB_1:def.8">PROB_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:103_3"/>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <font color="Maroon" title="c5">BSeq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:103"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:103"/>
for <font color="Olive" title="b1">E</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">S</font> where <font color="Olive" title="b2">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b2">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span>  holds <br/><font color="Olive" title="b1">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">S</font> where <font color="Olive" title="b2">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b2">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">E</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:103_4"/><i><font color="Green" title="E54">A9</font></i>: 
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:103_4_1"/><i><font color="Green" title="E55">A10</font></i>: 
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 ;<br/><a NAME="E2:103_4_1"/><b>then </b><i><font color="Green" title="E56">A11</font></i>: 
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:103_4"><i><font color="Green" title="E54">A9</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><a NAME="E3:103_4_1"/>
 ex <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:103_4_1"><i><font color="Green" title="E55">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E4:103_4_1"/>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:103_4_1"><i><font color="Green" title="E56">A11</font></i></a>, <a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:103_4"/>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:103"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">Y</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span>  as   <a href="topgen_4.html#NM1" title="TOPGEN_4:NM.1">SigmaField</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="topgen_4.html#E1:103"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="topgen_4.html#E3:103"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="topgen_4.html#E4:103"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="prob_1.html#T32" title="PROB_1:th.32">PROB_1:32</a>, <a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>
<a NAME="E7:103"/>
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="c1">T</font>  st <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5"><b>proof </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="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:103_5"/><i><font color="Green" title="E54">A12</font></i>: 
<font color="Maroon" title="c6">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:103_5"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">S</font> where <font color="Olive" title="b2">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b2">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span>  holds <br/><font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:103_5_1"/>
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">S</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:103_5_1"/><i><font color="Green" title="E55">A13</font></i>: 
( <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">S</font> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">S</font> &amp; <font color="Maroon" title="c8">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> )
 ;<br/>
<b>thus </b><a NAME="E4:103_5_1"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:103_5"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="topgen_4.html#E3:103_5_1"><i><font color="Green" title="E55">A13</font></i></a>, <a class="ref" href="topgen_4.html#D2" target="_self" title="TOPGEN_4:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:103_5"/>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:103"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">Y</font> = <font color="Maroon" title="c5">Y</font> as  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="topgen_4.html#D2" target="_self" title="TOPGEN_4:def.2">Def2</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">Y</font>
;<br/>

<div><i><font color="Green" title="E54">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:103_6"/>
<font color="Maroon" title="c7">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 ;<br/><a NAME="E2:103_6"/><b>then </b>
 ex <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c7">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> )
 ;<br/><b>hence </b><a NAME="E3:103_6"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E10:103"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font> &amp; <font color="Olive" title="b1">Z</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:103_7"/><i><font color="Green" title="E55">A15</font></i>: 
( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font> &amp; <font color="Maroon" title="c7">Z</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">Z</font> = <font color="Maroon" title="c7">Z</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E3:103_7"/>
<font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is  <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="topgen_4.html#E1:103_7"><i><font color="Green" title="E55">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:103_7"/>
<font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:103"/>
( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Y</font> &amp; ( for <font color="Olive" title="b1">Z</font> being <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font> holds <br/><font color="Maroon" title="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font> ) )
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:103"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="topgen_4.html#E9:103"><i><font color="Green" title="E54">A14</font></i></a>, <a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></i>;<br/>


</div>
