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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span>)</span> is <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> &amp;  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span>)</span> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> &amp;  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span>)</span> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> )</span><br/>

<b>set </b><font color="Maroon" title="c2">E</font> =  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span>)</span>;<br/>
<a NAME="E1:97"/>
 <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="prob_1.html#D14" title="PROB_1:def.14">PROB_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E2:97"/>
 <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span>)</span> is <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a>
 <b>by </b><i><a class="ref" href="topgen_4.html#T63" target="_self" title="TOPGEN_4:th.63">Th63</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span>)</span> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> &amp;  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span>)</span> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> )</span><br/>

<b>thus </b><a NAME="E3:97"/>
 <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span>)</span> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span>)</span> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a></span><br/>

<b>thus </b><a NAME="E4:97"/>
 <a href="prob_1.html#K10" title="PROB_1:func.10">sigma</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span>)</span> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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