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<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:63"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">Y</font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> holds  <font color="Olive">Y</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 ;<br/>

<a NAME="E2:63"/>
for <font color="Olive">X0</font> being non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> holds  <font color="Olive">X0</font> is <a href="tdlat_3.html#V4" target="_self">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">X0</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">A0</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A0</font>;<br/>
<a NAME="E2:63_1"/>
not  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A0</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T2">PCOMPS_1:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:63_1"/><i><font color="Green">E35</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T15">TSEP_1:15</a></i>;<br/>
<a NAME="E5:63_1"/>
<font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A0</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Y0</font> = <font color="Maroon">X0</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font> <b>by </b><i>, <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B0</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E8:63_1"/>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A0</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T47">PRE_TOPC:47</a></i>;<br/>
<a NAME="E9:63_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">B0</font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#D3">TOPS_1:def 3</a></i>;<br/>
<a NAME="E10:63_1"/>
<font color="Maroon">Y</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:63_1"/>
<font color="Maroon">X0</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T14" target="_self">Th14</a>, <a class="ref" href="tdlat_3.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:63"/>
<font color="Maroon">X</font> is <a href="tdlat_3.html#V5" target="_self">hereditarily_extremally_disconnected</a>
 <b>by </b><i/>;<br/>


</div>
