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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<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>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:67"/><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="E2:67"/><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>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">F</font> being  <a href="card_4.html#V1" title="CARD_4:attr.1">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="E4:67"/><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="topgen_4.html#E1:67"><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_4.html#V1" title="CARD_4:attr.1">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="E6:67"/><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="topgen_4.html#E2:67"><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#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"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></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"><b>suppose </b></a><a NAME="E1:67_1_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>  )
 ;<br/><div class="add"><a NAME="E2:67_1_1"/><b>then </b><i><font color="Green" title="E41">A5</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="topgen_4.html#E4:67"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="topgen_4.html#T28" target="_self" title="TOPGEN_4:th.28">Th28</a>, <a class="txt" href="topgen_4.html#E6:67"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/><a NAME="E3:67_1_1"/><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#T22" target="_self" title="TOPGEN_4:th.22">Th22</a></i>;<br/><a NAME="E4:67_1_1"/><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="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</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#T26" target="_self" title="TOPGEN_4:th.26">Th26</a></i>;<br/><a NAME="E5:67_1_1"/>
<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_4.html#V1" title="CARD_4:attr.1">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="E6:67_1_1"/><b>then </b>
<font color="Maroon" title="c6">H</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E4:67_1_1"><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="E7:67_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>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:67_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="topgen_4.html#E3:67_1_1"><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>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_2"><b>suppose </b></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> 
 ;<br/><div class="add"><b>hence </b><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>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:67"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="67_1_3"><b>suppose </b></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> 
 ;<br/><div class="add"><b>hence </b><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>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:67"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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