<?xml version="1.0"?>
<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/>

<a NAME="E1:74"/><i><font color="Green">E21</font></i>: 
 <a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK3" target="_self">D-Union</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="tdlat_1.html#NK5" target="_self">D-Meet</a> <font color="Maroon">X</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D4">TDLAT_1:def 4</a></i>;<br/>
<b>thus </b><a NAME="E2:74"/>
( <font color="Maroon">X</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a> implies  <a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">X</font> is   <a href="lattices.html#NM3">M_Lattice</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:74_1"/>
<font color="Maroon">X</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 ;<br/>

<a NAME="E2:74_1"/><b>then </b>
 <a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="tdlat_1.html#K12">Open_Domains_Lattice</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:74_1"/>
 <a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">X</font> is   <a href="lattices.html#NM3">M_Lattice</a>
 ;<br/>


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

<b>assume </b><a NAME="E3:74"/><i><font color="Green">E35</font></i>: 
 <a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">X</font> is   <a href="lattices.html#NM3">M_Lattice</a>
 ;<br/>

<b>assume </b><a NAME="E4:74"/>
not <font color="Maroon">X</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 ;<br/>

<b>then consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:74"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">D</font> is <a href="tops_1.html#V4">condensed</a>
 <b>and </b><br/><a NAME="E7:74"/><i><font color="Green">E46</font></i>: 
 <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">D</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="tdlat_3.html#D1" target="_self">Def1</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> =  <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">D</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/>
<a NAME="E8:74"/><i><font color="Green">E47</font></i>: 
(  <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">D</font></span>)</span> is <a href="tops_1.html#V6">open_condensed</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span> is <a href="tops_1.html#V5">closed_condensed</a> )
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T22">TDLAT_1:22</a>, <a class="ref" href="tdlat_1.html#T23">TDLAT_1:23</a></i>;<br/>
<a NAME="E9:74"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <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">D</font></span>)</span> is <a href="tops_1.html#V4">condensed</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span> is <a href="tops_1.html#V4">condensed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T106">TOPS_1:106</a>, <a class="ref" href="tops_1.html#T107">TOPS_1:107</a></i>;<br/>
<a NAME="E10:74"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V5">closed_condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T22">TDLAT_1:22</a></i>;<br/>
<a NAME="E11:74"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V6">open_condensed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T101">TOPS_1:101</a></i>;<br/>
<a NAME="E12:74"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T107">TOPS_1:107</a></i>;<br/>
<a NAME="E13:74"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">E</font> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">E</font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_3.html#T17" target="_self">Th17</a>, <a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/>
<a NAME="E15:74"/><i><font color="Green">E51</font></i>: 
 <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">D</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">E</font> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">E</font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E16:74"/><b>then </b><i><font color="Green">E62</font></i>: 
 <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">D</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> =  <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">D</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_3.html#T17" target="_self">Th17</a>, <a class="txt" href="tdlat_3.html#E6:74"><i><font color="Green">E45</font></i></a>, <a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/>
<a NAME="E18:74"/><i><font color="Green">E63</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">E</font> where <font color="Olive">E</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">E</font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E19:74"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> =  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_3.html#T17" target="_self">Th17</a>, <a class="txt" href="tdlat_3.html#E8:74"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/>
<a NAME="E21:74"/>
(  <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">D</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T18" target="_self">Th18</a>, <a class="ref" href="tops_1.html#D6">TOPS_1:def 6</a></i>;<br/>
<a NAME="E22:74"/><b>then </b>
 <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">D</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E23:74"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="txt" href="tdlat_3.html#E7:74"><i><font color="Green">E46</font></i></a>, <a class="txt" href="tdlat_3.html#E9:74"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tdlat_2.html#T89">TDLAT_2:89</a></i>;<br/>
<a NAME="E24:74"/><i><font color="Green">E80</font></i>: 
 <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">D</font></span>)</span> <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> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#D8">TOPS_1:def 8</a></i>;<br/>
<a NAME="E25:74"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E26:74"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E27:74"/><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font></span>)</span> <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">D</font></span>)</span>
 
;<br/>
<font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p0">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p5">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_3.html#T17" target="_self">Th17</a>, <a class="ref" href="tdlat_2.html#T87">TDLAT_2:87</a></i>
<br/>.= 
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
<b>by </b><i><a class="txt" href="tdlat_3.html#E15:74"><i><font color="Green">E51</font></i></a></i>
;<br/>
<b>then </b><i><font color="Green">E95</font></i>: <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_3.html#T17" target="_self">Th17</a>, , <a class="ref" href="tdlat_2.html#T87">TDLAT_2:87</a></i>
<br/>.= 
 <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">D</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E30:74"/><i><font color="Green">E96</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#D7">TOPS_1:def 7</a></i>;<br/>
<i><font color="Green">E97</font></i>: <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  = 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</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> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T50">PRE_TOPC:50</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_3.html#T20" target="_self">Th20</a>, <a class="ref" href="tdlat_1.html#T6">TDLAT_1:6</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K4">\/</a> <span class="p4">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T50">PRE_TOPC:50</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tops_1.html#T27">TOPS_1:27</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="tops_1.html#T43">TOPS_1:43</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="tops_1.html#T27">TOPS_1:27</a></i>
;<br/>
<b>then </b><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_3.html#T17" target="_self">Th17</a>, <a class="ref" href="tdlat_2.html#T87">TDLAT_2:87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tops_1.html#T43">TOPS_1:43</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p5">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_3.html#T17" target="_self">Th17</a>, <a class="ref" href="tdlat_2.html#T87">TDLAT_2:87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E35:74"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tdlat_3.html#T19" target="_self">Th19</a>, , , <a class="ref" href="lattices.html#D12">LATTICES:def 12</a></i>;<br/>


</div>
