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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:64"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="topgen_4.html#V5" target="_self">F_sigma</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being  <a href="card_4.html#V1">countable</a> <a href="tops_2.html#V2">closed</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:64"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="topgen_4.html#D6" target="_self">Def6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1"><b>suppose </b></a><a NAME="E1:64_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:64_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_2.html#T11">TOPS_2:11</a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E3:64_1_1"/>
 <a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">c<sub>3</sub></font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T16">TOPS_2:16</a></i>;<br/><b>hence </b><a NAME="E4:64_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="topgen_4.html#V6" target="_self">G_delta</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:64_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topgen_4.html#E3:64"><i><font color="Green">E37</font></i></a>, <a class="ref" href="topgen_4.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_2"><b>suppose </b></a><a NAME="E1:64_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:64_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E3:64"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/><a NAME="E3:64_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T27">PRE_TOPC:27</a></i>;<br/><b>hence </b><a NAME="E4:64_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="topgen_4.html#V6" target="_self">G_delta</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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