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<b>let </b><font color="Maroon">T</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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> is <a href="topgen_4.html#V6" target="_self">G_delta</a>
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">B</font> is <a href="topgen_4.html#V6" target="_self">G_delta</a>
 ;<br/>

<b>consider </b><font color="Maroon">F</font> being  <a href="card_4.html#V1">countable</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:68"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:68"><i><font color="Green">E37</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">G</font> being  <a href="card_4.html#V1">countable</a> <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:68"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">H</font> =  <a href="topgen_4.html#K3" target="_self">UNION</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1"/>
( ( <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</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="68_1_1"><b>suppose </b></a><a NAME="E1:68_1_1"/>
( <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:68_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K3" target="_self">UNION</a> <font color="Maroon">F</font>,<font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="setfam_1.html#T40">SETFAM_1:40</a></i>;<br/><a NAME="E3:68_1_1"/>
 <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K3" target="_self">UNION</a> <font color="Maroon">F</font>,<font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:68_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:68"><i><font color="Green">E39</font></i></a>, , , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E5:68_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">H</font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i/>;<br/><a NAME="E6:68_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">H</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">G</font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i/>;<br/><a NAME="E7:68_1_1"/>
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">G</font></span><a href="mcart_1.html#K12">:]</a></span> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i><a class="ref" href="card_4.html#T55">CARD_4:55</a></i>;<br/><a NAME="E8:68_1_1"/><b>then </b>
<font color="Maroon">H</font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i>, <a class="ref" href="waybel12.html#T2">WAYBEL12:2</a></i>;<br/><b>hence </b><a NAME="E9:68_1_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> is <a href="topgen_4.html#V6" target="_self">G_delta</a>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2"><b>suppose </b></a><a NAME="E1:68_1_2"/>
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:68_1_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:68"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:68_1_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> is <a href="topgen_4.html#V6" target="_self">G_delta</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:68"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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