<?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>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:98"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_2.html#V1" target="_self">domains-family</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</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>assume </b><a NAME="E2:98"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><a NAME="E3:98"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<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> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></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> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:98_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></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> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E2:98_1"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></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> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E1:98"><i><font color="Green">E61</font></i></a>, <a class="ref" href="tdlat_2.html#T70" target="_self">Th70</a></i>;<br/>
<a NAME="E3:98_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></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> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <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="E4:98_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></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> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <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>5</sub></font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></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> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></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">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/>
<a NAME="E6:98_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</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/>
<b>assume </b><a NAME="E2:98_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:98_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></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> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:98"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tdlat_2.html#T71" target="_self">Th71</a></i>;<br/>
<b>hence </b><a NAME="E4:98_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:98_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tdlat_2.html#T89" target="_self">Th89</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:98_1"/><i><font color="Green">E66</font></i>: 
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> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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>7</sub></font> = <font color="Maroon">c<sub>6</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="E2:98_1_2"/>
<font color="Maroon">c<sub>7</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="E3:98_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</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="E4:98_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
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>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V4">condensed</a> )
 
;<br/>
<b>assume </b><a NAME="E5:98_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E6:98_1_2"/>
for <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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>assume </b><a NAME="E2:98_1_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:98_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E1:98"><i><font color="Green">E61</font></i></a>, <a class="ref" href="tdlat_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:98_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</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="E5:98_1_2_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>8</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>10</sub></font> = <font color="Maroon">c<sub>8</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/>
<a NAME="E7:98_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:98"><i><font color="Green">E62</font></i></a>, <a class="txt" href="tdlat_2.html#E5:98_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="tdlat_2.html#E2:98_1_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E8:98_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E5:98_1_2_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="tdlat_2.html#T89" target="_self">Th89</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:98_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></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> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:98"><i><font color="Green">E62</font></i></a>, <a class="txt" href="tdlat_2.html#E1:98_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="tdlat_2.html#E4:98_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tdlat_2.html#T71" target="_self">Th71</a></i>;<br/>
<b>hence </b><a NAME="E8:98_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:98_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tdlat_2.html#T89" target="_self">Th89</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:98_1"/>
 <a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V4">complete</a>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T91" target="_self">Th91</a></i>;<br/>
<b>hence </b><a NAME="E9:98_1"/>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<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> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></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> <span class="p3">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E6:98_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="tdlat_2.html#E7:98_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>


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

<b>thus </b><a NAME="E4:98"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<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="98_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:98_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:98_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="E3:98_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> <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="E4:98_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 <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>5</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/>
<a NAME="E6:98_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:98_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:98_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E1:98_2"><i><font color="Green">E63</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:98_2"/><i><font color="Green">E66</font></i>: 
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> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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>7</sub></font> = <font color="Maroon">c<sub>6</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/>
<b>assume </b><a NAME="E2:98_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:98_2_2"/>
<font color="Maroon">c<sub>7</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>hence </b><a NAME="E4:98_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:98_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tdlat_2.html#T89" target="_self">Th89</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:98_2"/>
 <a href="tdlat_1.html#K4">Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V4">complete</a>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T91" target="_self">Th91</a></i>;<br/>
<b>hence </b><a NAME="E9:98_2"/>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<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="txt" href="tdlat_2.html#E6:98_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="tdlat_2.html#E7:98_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>


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


</div>
