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

<span class="kw">let </span><font color="Maroon" title="c1">Y</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</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="Red">::  thesis: </font></span></a><span class="hide">  <a href="tdlat_1.html#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">Y</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">S</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_1.html#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">Y</font> implies <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">Y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:69_1"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_1.html#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">Y</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c2">S</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> ;<br/><a NAME="E3:69_1"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">D</font> where <font color="Olive" title="b1">D</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> : <font color="Olive" title="b1">D</font> is  <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a>  </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="tdlat_1.html#D1" title="TDLAT_1:def.1">TDLAT_1:def 1</a></span>;<br/><a NAME="E4:69_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">D</font> is  <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a>  )
 ;<br/><a NAME="E5:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="ref" href="tdlat_3.html#T35" target="_self" title="TDLAT_3:th.35">Th35</a></span>;<br/><a NAME="E6:69_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> : <font color="Olive" title="b1">E</font> is  <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>  </span>}</span> 
 ;<br/><span class="kw">hence </span><a NAME="E7:69_1"/>
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tdlat_1.html#D9" title="TDLAT_1:def.9">TDLAT_1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A2</font></span>: 
 <a href="tdlat_1.html#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E3:69"/>
 <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tdlat_1.html#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tdlat_1.html#T35" title="TDLAT_1:th.35">TDLAT_1:35</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:69"/>
 <a href="tdlat_1.html#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E35">A2</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
