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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E1:115"/><i><font color="Green" title="E71">A1</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V3" title="TDLAT_2:attr.3">open-domains-family</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:115"/><i><font color="Green" title="E72">A2</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<b>thus </b><a NAME="E3:115"/>
( <font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</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="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:115_1"/><i><font color="Green" title="E73">A3</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c4">A</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font></span>)</span>;<br/>
<a NAME="E2:115_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font></span>)</span> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E1:115"><i><font color="Green" title="E71">A1</font></i></a>, <a class="ref" href="tdlat_2.html#T83" target="_self" title="TDLAT_2:th.83">Th83</a></i>;<br/>
<a NAME="E3:115_1"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">C</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E4:115_1"/><b>then </b><i><font color="Green" title="E74">A4</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font></span>)</span> <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">T</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D9" title="TDLAT_1:def.9">TDLAT_1:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">a</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T103" target="_self" title="TDLAT_2:th.103">Th103</a></i>;<br/>
<a NAME="E6:115_1"/><i><font color="Green" title="E75">A5</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="tdlat_2.html#T103" target="_self" title="TDLAT_2:th.103">Th103</a></i>;<br/>
<b>assume </b><a NAME="E2:115_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E3:115_1_1"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:115"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="tdlat_2.html#T85" target="_self" title="TDLAT_2:th.85">Th85</a></i>;<br/>
<b>hence </b><a NAME="E4:115_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:115_1"><i><font color="Green" title="E74">A4</font></i></a>, <a class="ref" href="tdlat_2.html#T106" target="_self" title="TDLAT_2:th.106">Th106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:115_1"/><i><font color="Green" title="E76">A6</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="tdlat_2.html#T103" target="_self" title="TDLAT_2:th.103">Th103</a></i>;<br/>
<a NAME="E2:115_1_2"/>
<font color="Maroon" title="c7">B</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">T</font>
 
;<br/>
<a NAME="E3:115_1_2"/><b>then </b>
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">C</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D9" title="TDLAT_1:def.9">TDLAT_1:def 9</a></i>;<br/>
<a NAME="E4:115_1_2"/><b>then </b><i><font color="Green" title="E77">A7</font></i>: 
 ex <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> &amp; <font color="Olive" title="b1">C</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> )
 
;<br/>
<b>assume </b><a NAME="E5:115_1_2"/><i><font color="Green" title="E78">A8</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E6:115_1_2"/>
for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/><font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">C1</font> = <font color="Maroon" title="c8">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>assume </b><a NAME="E2:115_1_2_1"/><i><font color="Green" title="E79">A9</font></i>: 
<font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<a NAME="E3:115_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">C1</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E1:115"><i><font color="Green" title="E71">A1</font></i></a>, <a class="ref" href="tdlat_2.html#D4" target="_self" title="TDLAT_2:def.4">Def4</a></i>;<br/>
<a NAME="E4:115_1_2_1"/><b>then </b>
<font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">P</font> where <font color="Olive" title="b1">P</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">P</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E5:115_1_2_1"/><b>then </b><i><font color="Green" title="E80">A10</font></i>: 
<font color="Maroon" title="c8">C</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">T</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D9" title="TDLAT_1:def.9">TDLAT_1:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c8">C</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T103" target="_self" title="TDLAT_2:th.103">Th103</a></i>;<br/>
<a NAME="E7:115_1_2_1"/>
<font color="Maroon" title="c6">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:115"><i><font color="Green" title="E72">A2</font></i></a>, <a class="txt" href="tdlat_2.html#E5:115_1_2"><i><font color="Green" title="E78">A8</font></i></a>, <a class="txt" href="tdlat_2.html#E2:115_1_2_1"><i><font color="Green" title="E79">A9</font></i></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E8:115_1_2_1"/>
<font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E5:115_1_2_1"><i><font color="Green" title="E80">A10</font></i></a>, <a class="ref" href="tdlat_2.html#T106" target="_self" title="TDLAT_2:th.106">Th106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:115_1_2"/><b>then </b>
<font color="Maroon" title="c7">B</font> <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="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:115"><i><font color="Green" title="E72">A2</font></i></a>, <a class="txt" href="tdlat_2.html#E1:115_1"><i><font color="Green" title="E73">A3</font></i></a>, <a class="txt" href="tdlat_2.html#E4:115_1_2"><i><font color="Green" title="E77">A7</font></i></a>, <a class="ref" href="tdlat_2.html#T85" target="_self" title="TDLAT_2:th.85">Th85</a></i>;<br/>
<b>hence </b><a NAME="E8:115_1_2"/>
<font color="Maroon" title="c6">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:115_1"><i><font color="Green" title="E74">A4</font></i></a>, <a class="ref" href="tdlat_2.html#T106" target="_self" title="TDLAT_2:th.106">Th106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:115_1"/>
 <a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font> is <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T108" target="_self" title="TDLAT_2:th.108">Th108</a></i>;<br/>
<b>hence </b><a NAME="E9:115_1"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</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="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E6:115_1"><i><font color="Green" title="E75">A5</font></i></a>, <a class="txt" href="tdlat_2.html#E7:115_1"><i><font color="Green" title="E76">A6</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>


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

<b>thus </b><a NAME="E4:115"/>
( <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:115_2"/><i><font color="Green" title="E73">A11</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c4">A</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E2:115_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#T21" title="TDLAT_1:th.21">TDLAT_1:21</a></i>;<br/>
<a NAME="E3:115_2"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">C</font> where <font color="Olive" title="b1">C</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">C</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E4:115_2"/><b>then </b><i><font color="Green" title="E74">A12</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</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">T</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D9" title="TDLAT_1:def.9">TDLAT_1:def 9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">a</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T103" target="_self" title="TDLAT_2:th.103">Th103</a></i>;<br/>
<a NAME="E6:115_2"/><i><font color="Green" title="E75">A13</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:115_2_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:115_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E1:115_2"><i><font color="Green" title="E73">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:115_2"/><i><font color="Green" title="E76">A14</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="tdlat_2.html#T103" target="_self" title="TDLAT_2:th.103">Th103</a></i>;<br/>
<b>assume </b><a NAME="E2:115_2_2"/>
<font color="Maroon" title="c6">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E3:115_2_2"/>
<font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T14" title="PRE_TOPC:th.14">PRE_TOPC:14</a></i>;<br/>
<b>hence </b><a NAME="E4:115_2_2"/>
<font color="Maroon" title="c6">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:115_2"><i><font color="Green" title="E74">A12</font></i></a>, <a class="ref" href="tdlat_2.html#T106" target="_self" title="TDLAT_2:th.106">Th106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:115_2"/>
 <a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font> is <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T108" target="_self" title="TDLAT_2:th.108">Th108</a></i>;<br/>
<b>hence </b><a NAME="E9:115_2"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c3">X</font>,<span class="p1">(<span class="default"><a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E6:115_2"><i><font color="Green" title="E75">A13</font></i></a>, <a class="txt" href="tdlat_2.html#E7:115_2"><i><font color="Green" title="E76">A14</font></i></a>, <a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>


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


</div>
