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<div class="add">

<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"> 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> holds <br/> ( <font color="Olive" title="b1">F</font> is <a href="topgen_4.html#V2" title="TOPGEN_4:attr.2">all-closed-containing</a> &amp; <font color="Olive" title="b1">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> iff ( <font color="Olive" title="b1">F</font> is <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> &amp; <font color="Olive" title="b1">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> ) )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V2" title="TOPGEN_4:attr.2">all-closed-containing</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> iff ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> ) )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> implies ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V2" title="TOPGEN_4:attr.2">all-closed-containing</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> ) )</span>
<div class="add"><b>assume </b><a NAME="E1:29_1"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V2" title="TOPGEN_4:attr.2">all-closed-containing</a> &amp; <font color="Maroon" title="c2">F</font> 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"> ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> )</span><br/><a NAME="E2:29_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> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <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"> ( <font color="Maroon" title="c3">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<b>assume </b><a NAME="E1:29_1_1"/>
<font color="Maroon" title="c3">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="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<a NAME="E2:29_1_1"/><b>then </b>
<font color="Maroon" title="c3">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> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E1:29_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="topgen_4.html#D3" target="_self" title="TOPGEN_4:def.3">Def3</a></i>;<br/>
<a NAME="E3:29_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E1:29_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</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><b>hence </b><a NAME="E3:29_1"/>
( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> )
 <b>by </b><i><a class="txt" href="#E1:29_1"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="topgen_4.html#D2" target="_self" title="TOPGEN_4:def.2">Def2</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>
<b>end;</b></div>

<b>assume </b><a NAME="E2:29"/><i><font color="Green" title="E15">A2</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V1" title="TOPGEN_4:attr.1">all-open-containing</a> &amp; <font color="Maroon" title="c2">F</font> 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"> ( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V2" title="TOPGEN_4:attr.2">all-closed-containing</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> )</span><br/>

<a NAME="E3:29"/>
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> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> holds <br/><font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <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"> ( <font color="Maroon" title="c3">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> implies <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<b>assume </b><a NAME="E1:29_2"/>
<font color="Maroon" title="c3">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<a NAME="E2:29_2"/><b>then </b>
<font color="Maroon" title="c3">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> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="topgen_4.html#D2" target="_self" title="TOPGEN_4:def.2">Def2</a></i>;<br/>
<a NAME="E3:29_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:29_2"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</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>
<b>hence </b><a NAME="E4:29"/>
( <font color="Maroon" title="c2">F</font> is <a href="topgen_4.html#V2" title="TOPGEN_4:attr.2">all-closed-containing</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="prob_1.html#V1" title="PROB_1:attr.1">compl-closed</a> )
 <b>by </b><i><a class="txt" href="#E2:29"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="topgen_4.html#D3" target="_self" title="TOPGEN_4:def.3">Def3</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>
