<?xml version="1.0"?>
<div class="add">

<b>thus </b><a NAME="E1:91_1_1"/>
not  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> is <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> &amp;  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> &amp;  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> )</span><br/>

<b>thus </b><a NAME="E2:91_1_1"/>
 <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> is <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> &amp;  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="topgen_4.html#D2" target="_self" title="TOPGEN_4:def.2">TOPGEN_4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_1"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font></span><br/>

<b>thus </b><a NAME="E2:91_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<a NAME="E3:91_1_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> holds <br/><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font>
 
;<br/>
<b>hence </b><a NAME="E4:91_1_1"/>
 <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a>
 <b>by </b><i><a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a></span><br/>

<a NAME="E5:91_1_1"/>
for <font color="Olive" title="b1">G</font> being <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a> <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">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> holds <br/> <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font>
 
;<br/>
<b>hence </b><a NAME="E6:91_1_1"/>
 <a href="topgen_4.html#K1" title="TOPGEN_4:func.1">TotFam</a> <font color="Maroon" title="c1">T</font> is <a href="topgen_4.html#V3" title="TOPGEN_4:attr.3">closed_for_countable_unions</a>
 <b>by </b><i><a class="ref" href="topgen_4.html#D4" target="_self" title="TOPGEN_4:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
