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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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:69"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="topgen_4.html#V6" target="_self">G_delta</a>
 <b>and </b><br/><a NAME="E2:69"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="topgen_4.html#V6" target="_self">G_delta</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:69"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:69"><i><font color="Green">E37</font></i></a>, <a class="ref" href="topgen_4.html#D7" target="_self">Def7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:69"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E2:69"><i><font color="Green">E38</font></i></a>, <a class="ref" href="topgen_4.html#D7" target="_self">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="topgen_4.html#K2" target="_self">INTERSECTION</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>5</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>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>5</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="69_1_1"><b>suppose </b></a><a NAME="E1:69_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>5</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="E2:69_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E4:69"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topgen_4.html#E6:69"><i><font color="Green">E40</font></i></a>, <a class="ref" href="topgen_4.html#T34" target="_self">Th34</a></i>;<br/><a NAME="E3:69_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i><a class="ref" href="topgen_4.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E4:69_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="ref" href="topgen_4.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E5:69_1_1"/>
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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="E6:69_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i><a class="txt" href="topgen_4.html#E4:69_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel12.html#T2">WAYBEL12:2</a></i>;<br/><b>hence </b><a NAME="E7:69_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="topgen_4.html#V6" target="_self">G_delta</a>
 <b>by </b><i><a class="ref" href="topgen_4.html#D7" target="_self">Def7</a>, <a class="txt" href="topgen_4.html#E3:69_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topgen_4.html#E2:69_1_1"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_2"><b>suppose </b></a><a NAME="E1:69_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:69_1_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</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>  )
 <b>by </b><i><a class="txt" href="topgen_4.html#E4:69"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topgen_4.html#E6:69"><i><font color="Green">E40</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E3:69_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>hence </b><a NAME="E4:69_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> 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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