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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E1:53"/>
<font color="Maroon" title="c2">F</font> is <a href="topgen_1.html#V4" title="TOPGEN_1:attr.4">dense-in-itself</a>
 ;<br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green" title="E32">A1</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topgen_1.html#T39" target="_self" title="TOPGEN_1:th.39">Th39</a></i>;<br/>
<a NAME="E3:53"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K3" title="TOPGEN_1:func.3">Der</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_1.html#K2" title="TOPGEN_1:func.2">Der</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topgen_1.html#T36" target="_self" title="TOPGEN_1:th.36">Th36</a></i>;<br/>
<a NAME="E4:53"/><b>then </b>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_1.html#K2" title="TOPGEN_1:func.2">Der</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E2:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E5:53"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">F</font> is <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">dense-in-itself</a>
 <b>by </b><i><a class="ref" href="topgen_1.html#D7" target="_self" title="TOPGEN_1:def.7">Def7</a></i>;<br/>


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