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

<span class="kw">let </span><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>; <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">X</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  ex <font color="Olive" title="b2">Y</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 <br/>( <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> &amp; ( for <font color="Olive" title="b3">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="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Z</font> holds <br/><font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Z</font> ) )</span><br/><span class="kw">let </span><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>; <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">Y</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 <br/>( <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> &amp; ( for <font color="Olive" title="b2">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="b2">Z</font> holds <br/><font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Z</font> ) )</span><br/>



<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<div><span class="lab"><font color="Green" title="E51">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">Z</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="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>  implies <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Z</font> )</span><br/><span class="kw">assume </span><a NAME="E1:105_1"/>
<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> 
 ; <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="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Z</font></span><br/><a NAME="E2:105_1"/><span class="kw">then </span>
 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="c5">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/><span class="kw">hence </span><a NAME="E3:105_1"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Z</font>
 ; <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="E2:105"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="105_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><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="E1:105_2"/>
 <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/>
<span class="kw">hence </span><a NAME="E2:105_2"/>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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> 
 ; <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="E3:105"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
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="105_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <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="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>  implies <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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:105_3"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<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> 
 ; <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="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> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">Z</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="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>  implies <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> )</span><br/><span class="kw">assume </span><a NAME="E1:105_3_1"/>
<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> 
 ; <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="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></span><br/><a NAME="E2:105_3_1"/><span class="kw">then </span>
( <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> &amp;  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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_3"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:105_3_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1: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">hence </span><a NAME="E3:105_3"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1: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="E4:105"/><span class="lab"><font color="Green" title="E54">A5</font></span>: 
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  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Olive" title="b1">BSeq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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/> <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="105_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c5">BSeq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>  implies  <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>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:105_4"/><span class="lab"><font color="Green" title="E55">A6</font></span>: 
 <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c5">BSeq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">Z</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="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>  implies  <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> )</span><br/><span class="kw">assume </span><a NAME="E1:105_4_1"/><span class="lab"><font color="Green" title="E56">A7</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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></span><br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:105_4_1"/><span class="lab"><font color="Green" title="E57">A8</font></span>: 
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font>
 <span class="kw">and </span><br/><a NAME="E4:105_4_1"/>
<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>
 <span class="kw">and </span><br/><a NAME="E5:105_4_1"/><span class="lab"><font color="Green" title="E58">A9</font></span>: 
<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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_4_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c5">BSeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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></span><br/><a NAME="E1:105_4_1_1"/>
<font color="Maroon" title="c5">BSeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c5">BSeq</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T52" title="NAT_1:th.52">NAT_1:52</a></span>;<br/><span class="kw">hence </span><a NAME="E2:105_4_1_1"/>
<font color="Maroon" title="c5">BSeq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_4"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="txt" href="#E1:105_4_1"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1: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="E7:105_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A10</font></span>: 
 <a href="topgen_3.html#K1" title="TOPGEN_3:func.1">rng</a> <font color="Maroon" title="c5">BSeq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></span>;<br/><a NAME="E8:105_4_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:105_4_1"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="ref" href="topgen_4.html#T13" target="_self" title="TOPGEN_4:th.13">Th13</a></span>;<br/><span class="kw">hence </span><a NAME="E9:105_4_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:105_4_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="txt" href="#E7:105_4_1"><span class="lab"><font color="Green" title="E59">A10</font></span></a>, <a class="ref" href="prob_1.html#D8" title="PROB_1:def.8">PROB_1:def 8</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="E3:105_4"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1: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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">Z</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="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>  implies  <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> )</span><br/><span class="kw">assume </span><a NAME="E1:105_5"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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></span><br/><a NAME="E2:105_5"/><span class="kw">then </span>
 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="c5">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/><a NAME="E3:105_5"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_4.html#T13" target="_self" title="TOPGEN_4:th.13">Th13</a></span>;<br/><span class="kw">hence </span><a NAME="E4:105_5"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T10" title="PROB_1:th.10">PROB_1:10</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="E6:105"/><span class="kw">then </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#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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E4:105"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E3:105"><span class="lab"><font color="Green" title="E53">A3</font></span></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></span>;<br/>
<a NAME="E8:105"/>
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="105_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <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="c6">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:105_6"/><span class="lab"><font color="Green" title="E55">A11</font></span>: 
<font color="Maroon" title="c6">A</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="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font></span><br/>

<a NAME="E2:105_6"/>
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="105_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">Y</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="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>  implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:105_6_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> 
 ; <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="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font></span><br/>

<a NAME="E2:105_6_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c7">Y</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/>
<span class="kw">hence </span><a NAME="E3:105_6_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_6"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="topgen_4.html#D2" target="_self" title="TOPGEN_4:def.2">Def2</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="E3:105_6"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1: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">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_4.html#D2" target="_self" title="TOPGEN_4:def.2">Def2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">Y</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="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> ) )</span><br/>

<a NAME="E10:105"/>
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="105_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">Z</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="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> implies <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> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:105_7"/><span class="lab"><font color="Green" title="E55">A12</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:105_7"/><span class="lab"><font color="Green" title="E56">A13</font></span>: 
<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>
 ; <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="c6">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Z</font></span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105_7"><span class="lab"><font color="Green" title="E56">A13</font></span></a></span>;<br/>
<a NAME="E4:105_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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:105_7"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="txt" href="#E2:105_7"><span class="lab"><font color="Green" title="E56">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:105_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</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="E11:105"/>
( <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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:105"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:105"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</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>
