<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> holds <br/> ( ( <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="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#V5" title="TOPS_1:attr.5">closed_condensed</a>  implies <font color="Olive" title="b1">A</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>  ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:54"/>
( <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> holds <br/> ( ( <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="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#V5" title="TOPS_1:attr.5">closed_condensed</a>  implies <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> holds <br/> ( ( <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="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#V5" title="TOPS_1:attr.5">closed_condensed</a>  implies <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="54_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:54_1"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> holds <br/> ( ( <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="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#V5" title="TOPS_1:attr.5">closed_condensed</a>  implies <font color="Olive" title="b1">A</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>  ) )</span><br/>

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_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#V5" title="TOPS_1:attr.5">closed_condensed</a>  implies <font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:54_1"/>
( <font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_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>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>  implies <font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:54_1_1"/>
<font color="Maroon" title="c2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> </span><br/>

<a NAME="E2:54_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T67" title="TOPS_1:th.67">TOPS_1:67</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:54_1_1"/>
<font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="tdlat_3.html#T33" target="_self" title="TDLAT_3:th.33">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:54_1"/>
( <font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>  implies <font color="Maroon" title="c2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:54_1_2"/>
<font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="E2:54_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T66" title="TOPS_1:th.66">TOPS_1:66</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:54_1_2"/>
<font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="tdlat_3.html#T33" target="_self" title="TDLAT_3:th.33">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:54"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
 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> holds <br/> ( ( <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="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#V5" title="TOPS_1:attr.5">closed_condensed</a>  implies <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E3:54"/>
 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/> <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:54_2"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span></span><br/>

<a NAME="E2:54_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c2">A</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> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#D6" title="TOPS_1:def.6">TOPS_1:def 6</a></span>;<br/>
<a NAME="E3:54_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span> is  <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tdlat_1.html#T22" title="TDLAT_1:th.22">TDLAT_1:22</a></span>;<br/>
<a NAME="E4:54_2"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/>
<a NAME="E5:54_2"/><span class="kw">then </span>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#D8" title="TOPS_1:def.8">TOPS_1:def 8</a></span>;<br/>
<a NAME="E6:54_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tdlat_2.html#T1" title="TDLAT_2:th.1">TDLAT_2:1</a></span>;<br/>
<a NAME="E7:54_2"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="tops_1.html#D6" title="TOPS_1:def.6">TOPS_1:def 6</a></span>;<br/>
<a NAME="E8:54_2"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_2"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:54_2"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54_2"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:54"/>
<font color="Maroon" title="c1">X</font> is  <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tdlat_3.html#T34" target="_self" title="TDLAT_3:th.34">Th36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
