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

<b>let </b><font color="Maroon">c<sub>1</sub></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="E1:93"/>
 <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<a NAME="E1:93_1"/>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T14">TDLAT_1:14</a></i>;<br/>
<a NAME="E2:93_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:93_1"/><b>then </b>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> 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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T86" target="_self">Th86</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tdlat_2.html#E2:93_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/>
<a NAME="E6:93_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <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">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tdlat_2.html#T86" target="_self">Th86</a></i>;<br/>
<a NAME="E2:93_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E3:93_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/>
<a NAME="E4:93_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> )
 
;<br/>
<a NAME="E5:93_1_1"/><b>then </b><i><font color="Green">E57</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">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#D6">TOPS_1:def 6</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>4</sub></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#K4">Cl</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_2.html#T87" target="_self">Th87</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="tdlat_2.html#E5:93_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:93_1"/>
 <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="lattice2.html#T21">LATTICE2:21</a></i>;<br/>


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

<b>thus </b><a NAME="E2:93"/>
 <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><b>proof </b></a><div class="add">

<a NAME="E1:93_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T15">TDLAT_1:15</a></i>;<br/>
<a NAME="E2:93_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:93_2"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> 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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T86" target="_self">Th86</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tdlat_2.html#E2:93_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/>
<a NAME="E6:93_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <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">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <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">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tdlat_2.html#T86" target="_self">Th86</a></i>;<br/>
<a NAME="E2:93_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E3:93_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D1">TDLAT_1:def 1</a></i>;<br/>
<a NAME="E4:93_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> )
 
;<br/>
<a NAME="E5:93_2_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#D6">TOPS_1:def 6</a></i>;<br/>
<a NAME="E6:93_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T14">PRE_TOPC:14</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>4</sub></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"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="tdlat_2.html#T87" target="_self">Th87</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> <font color="Maroon">c<sub>5</sub></font></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">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tdlat_2.html#E6:93_2_1"><i><font color="Green">E58</font></i></a>, <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> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="tdlat_2.html#E6:93_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="tdlat_2.html#E5:93_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:93_2"/>
 <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="lattice2.html#T24">LATTICE2:24</a></i>;<br/>


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


</div>
