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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is  <a href="topgen_1.html#V4" title="TOPGEN_1:attr.4">dense-in-itself</a>  holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">F</font> is  <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">dense-in-itself</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is  <a href="topgen_1.html#V4" title="TOPGEN_1:attr.4">dense-in-itself</a>  implies  <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>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:55"/>
<font color="Maroon" title="c2">F</font> is  <a href="topgen_1.html#V4" title="TOPGEN_1:attr.4">dense-in-itself</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<a NAME="E2:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_1.html#T39" target="_self" title="TOPGEN_1:th.39">Th39</a></span>;<br/>
<a NAME="E3:55"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_1.html#T36" target="_self" title="TOPGEN_1:th.36">Th36</a></span>;<br/>
<a NAME="E4:55"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:55"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_1.html#D7" target="_self" title="TOPGEN_1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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