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<span class="kw">thus </span><a NAME="E1:92_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E2:92_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="92_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:92_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E2:92_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E3:92_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/>
<span class="kw">hence </span><a NAME="E4:92_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E5:92_1_1"/>
for <font color="Olive" title="b1">G</font> being  <a href="card_3.html#V4" title="CARD_3:attr.4">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/>
<span class="kw">hence </span><a NAME="E6:92_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topgen_4.html#D4" target="_self" title="TOPGEN_4:def.4">Def4</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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