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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">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>





<b>assume </b><a NAME="E1:46"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" target="_self">Der</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K3" target="_self">Der</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="topgen_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:46"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="topgen_1.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E7:46"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a>, <a class="ref" href="topgen_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E8:46"/><b>then </b>
 <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E1"><i><font color="Green">Lemma24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:46"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="topgen_1.html#T4" target="_self">Th4</a>, <a class="ref" href="topgen_1.html#T5" target="_self">Th5</a></i>;<br/>


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