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<b>let </b><font color="Maroon" title="c1">X</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"> ( <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a> iff 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">X</font>  st <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> holds <br/>( <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> &amp; <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> ) )</span><br/>

<b>thus </b><a NAME="E1:51"/>
( <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a> implies 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">X</font>  st <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> holds <br/>( <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> &amp; <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</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">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> holds <br/>( <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> &amp; <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> ) ) implies <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</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">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> holds <br/>( <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> &amp; <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> )</span><br/>

<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">X</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">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> implies ( <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> ) )</span><br/>

<b>assume </b><a NAME="E2:51_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</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="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> )</span><br/>

<a NAME="E3:51_1"/><b>then </b>
( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="#E1:51_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="tdlat_3.html#T34" target="_self" title="TDLAT_3:th.34">Th34</a></i>;<br/>
<b>hence </b><a NAME="E4:51_1"/>
( <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> )
 <b>by </b><i><a class="txt" href="#E2:51_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="tops_1.html#T106" title="TOPS_1:th.106">TOPS_1:106</a>, <a class="ref" href="tops_1.html#T107" title="TOPS_1:th.107">TOPS_1:107</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="E2:51"/><i><font color="Green" title="E27">A3</font></i>: 
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">X</font>  st <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> holds <br/>( <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> &amp; <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </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">X</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">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> implies ( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) )</span><br/><b>assume </b><a NAME="E1:51_2"/>
<font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</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#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><br/><a NAME="E2:51_2"/><b>then </b>
( <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> )
 <b>by </b><i><a class="txt" href="#E2:51"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:51_2"/>
( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="tops_1.html#T106" title="TOPS_1:th.106">TOPS_1:106</a>, <a class="ref" href="tops_1.html#T107" title="TOPS_1:th.107">TOPS_1:107</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="E4:51"/>
<font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T34" target="_self" title="TDLAT_3:th.34">Th34</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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