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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>thus </b><a NAME="E1:46"/>
( <font color="Maroon">X</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a> implies for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> holds <br/> <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">X</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:46_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<b>assume </b><a NAME="E3:46_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E4:46_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T10" target="_self">Th10</a>, <a class="ref" href="tdlat_3.html#T11" target="_self">Th11</a>, <a class="txt" href="tdlat_3.html#E1:46_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tsep_1.html#T40">TSEP_1:40</a></i>;<br/>
<b>hence </b><a NAME="E5:46_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T40">TSEP_1:40</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:46"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> holds <br/> <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">B</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:46_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:46_2"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E3:46_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/><a NAME="E4:46_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T10" target="_self">Th10</a>, <a class="ref" href="tdlat_3.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E5:46_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><a NAME="E6:46_2"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a>, <a class="ref" href="tops_1.html#D1">TOPS_1:def 1</a></i>;<br/><a NAME="E7:46_2"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E8:46_2"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:46"/>
<font color="Maroon">X</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 <b>by </b><i/>;<br/>


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