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<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a> &amp; <font color="Olive" title="b2">B</font> is <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a> holds <br/><font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font> is <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a> implies <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:66"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a>
 <b>and </b><br/><a NAME="E2:66"/><i><font color="Green" title="E38">A2</font></i>: 
<font color="Maroon" title="c3">B</font> is <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a></span><br/>

<b>consider </b><font color="Maroon" title="c4">F</font> being  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> <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="E4:66"/><i><font color="Green" title="E39">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="topgen_4.html#D6" target="_self" title="TOPGEN_4:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">G</font> being  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> <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="E6:66"/><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="#E2:66"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="topgen_4.html#D6" target="_self" title="TOPGEN_4:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">H</font> =  <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</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="E8:66"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="txt" href="#E4:66"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="#E6:66"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="setfam_1.html#T39" title="SETFAM_1:th.39">SETFAM_1:39</a></i>;<br/>
<a NAME="E9:66"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c6">H</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 
<b>by </b><i><a class="ref" href="topgen_4.html#T21" target="_self" title="TOPGEN_4:th.21">Th21</a></i>;<br/>
<a NAME="E10:66"/><i><font color="Green" title="E43">A7</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c6">H</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="topgen_4.html#T25" target="_self" title="TOPGEN_4:th.25">Th25</a></i>;<br/>
<a NAME="E11:66"/>
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> is <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T55" title="CARD_4:th.55">CARD_4:55</a></i>;<br/>
<a NAME="E12:66"/><b>then </b>
<font color="Maroon" title="c6">H</font> is <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>
 
<b>by </b><i><a class="txt" href="#E10:66"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="waybel12.html#T2" title="WAYBEL12:th.2">WAYBEL12:2</a></i>;<br/>
<b>hence </b><a NAME="E13:66"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a>
 <b>by </b><i><a class="txt" href="#E8:66"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="#E9:66"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="topgen_4.html#D6" target="_self" title="TOPGEN_4:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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