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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"> 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="tdlat_2.html#V1" title="TDLAT_2:attr.1">domains-family</a> holds <br/> <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Olive" title="b1">F</font> is <a href="tdlat_2.html#V2" title="TDLAT_2:attr.2">closed-domains-family</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="tdlat_2.html#V1" title="TDLAT_2:attr.1">domains-family</a> implies  <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V2" title="TDLAT_2:attr.2">closed-domains-family</a> )</span><br/>



<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V1" title="TDLAT_2:attr.1">domains-family</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V2" title="TDLAT_2:attr.2">closed-domains-family</a></span><br/>

<a NAME="E2:79"/>
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="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Olive" title="b1">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font> implies <font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> )</span><br/>

<b>assume </b><a NAME="E1:79_1"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</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"> <font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a></span><br/>

<b>then consider </b><font color="Maroon" title="c4">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:79_1"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c4">P</font>
 <b>and </b><br/><a NAME="E4:79_1"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D3" title="PCOMPS_1:def.3">PCOMPS_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">P</font> = <font color="Maroon" title="c4">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E6:79_1"/>
<font color="Maroon" title="c5">P</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E1:79"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="tdlat_2.html#E4:79_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="tdlat_2.html#D2" target="_self" title="TDLAT_2:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E7:79_1"/>
<font color="Maroon" title="c3">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E3:79_1"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="tdlat_1.html#T24" title="TDLAT_1:th.24">TDLAT_1:24</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>hence </b><a NAME="E3:79"/>
 <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V2" title="TDLAT_2:attr.2">closed-domains-family</a>
 <b>by </b><i><a class="ref" href="tdlat_2.html#D3" target="_self" title="TDLAT_2: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/>


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