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<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">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#K4" title="SUBSET_1:func.4">\/</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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K4" title="SUBSET_1:func.4">\/</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/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a> 
 <span class="kw">and </span><br/><a NAME="E2:67"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:67"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="topgen_4.html#D6" target="_self" title="TOPGEN_4:def.6">Def6</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:67"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="topgen_4.html#D6" target="_self" title="TOPGEN_4:def.6">Def6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">H</font> =  <a href="topgen_4.html#K3" title="TOPGEN_4:func.3">UNION</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:67_1"/>
( ( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_1"><span class="kw">suppose </span></a><a NAME="E1:67_1_1"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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/><div class="add"><a NAME="E2:67_1_1"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
<font color="Maroon" title="c6">H</font> is  <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_4.html#T22" target="_self" title="TOPGEN_4:th.22">Th22</a></span>;<br/><a NAME="E3:67_1_1"/>
(  <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> &amp; <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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_4.html#T26" target="_self" title="TOPGEN_4:th.26">Th26</a>, <a class="ref" href="card_4.html#T55" title="CARD_4:th.55">CARD_4:55</a></span>;<br/><a NAME="E4:67_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<font color="Maroon" title="c6">H</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel12.html#T2" title="WAYBEL12:th.2">WAYBEL12:2</a></span>;<br/><a NAME="E5:67_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:67"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E1:67_1_1"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="topgen_4.html#T28" target="_self" title="TOPGEN_4:th.28">Th28</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:67_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is  <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1_1"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E4:67_1_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="topgen_4.html#D6" target="_self" title="TOPGEN_4:def.6">Def6</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="hsNdiv(this)" href="javascript:()" title="67_1_2"><span class="kw">suppose </span></a><a NAME="E1:67_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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/><div class="add"><span class="kw">hence </span><a NAME="E2:67_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is  <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E38">A2</font></span></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="hsNdiv(this)" href="javascript:()" title="67_1_3"><span class="kw">suppose </span></a><a NAME="E1:67_1_3"/>
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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/><div class="add"><span class="kw">hence </span><a NAME="E2:67_1_3"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is  <a href="topgen_4.html#V5" title="TOPGEN_4:attr.5">F_sigma</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E37">A1</font></span></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><span class="kw">end;</span></div>

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