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<div class="add">

<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"> ( ( for <font color="Olive" title="b1">X0</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">X0</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a> ) implies <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V5" title="TDLAT_3:attr.5">hereditarily_extremally_disconnected</a> )</span><br/>

<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E31">A1</font></i>: 
for <font color="Olive" title="b1">Y</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">Y</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"> <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V5" title="TDLAT_3:attr.5">hereditarily_extremally_disconnected</a></span><br/>

<a NAME="E2:63"/>
for <font color="Olive" title="b1">X0</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">X0</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">X0</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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">X0</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c3">A0</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">A</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A0</font>;<br/>
<a NAME="E2:63_1"/>
not  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A0</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T2" title="PCOMPS_1:th.2">PCOMPS_1:2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">Y</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</a> <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:63_1"/><i><font color="Green" title="E32">A2</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T15" title="TSEP_1:th.15">TSEP_1:15</a></i>;<br/>
<a NAME="E5:63_1"/>
<font color="Maroon" title="c3">A0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A0</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">Y0</font> = <font color="Maroon" title="c2">X0</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c5">Y</font> <b>by </b><i><a class="txt" href="#E4:63_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="tsep_1.html#T4" title="TSEP_1:th.4">TSEP_1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">B0</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">Y0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">Y</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<a NAME="E8:63_1"/>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">B0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A0</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">Y</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">A0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="txt" href="#E4:63_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="pre_topc.html#T47" title="PRE_TOPC:th.47">PRE_TOPC:47</a></i>;<br/>
<a NAME="E9:63_1"/><b>then </b><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c7">B0</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#D3" title="TOPS_1:def.3">TOPS_1:def 3</a></i>;<br/>
<a NAME="E10:63_1"/>
<font color="Maroon" title="c5">Y</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a>
 
<b>by </b><i><a class="txt" href="#E1:63"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:63_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a>
 <b>by </b><i><a class="txt" href="#E9:63_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="tdlat_3.html#T39" target="_self" title="TDLAT_3:th.39">Th39</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="E3:63"/>
<font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V5" title="TDLAT_3:attr.5">hereditarily_extremally_disconnected</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#D5" target="_self" title="TDLAT_3:def.5">Def5</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>
