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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">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#K4" title="TDLAT_1:func.4">Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span>;<br/>



<a NAME="E1:95"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K4" title="TDLAT_1:func.4">Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span>
 
;<br/>
<a NAME="E2:95"/><b>then </b><i><font color="Green" title="E58">A1</font></i>: 
<font color="Maroon" title="c2">X</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">T</font>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T86" target="_self" title="TDLAT_2:th.86">Th86</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">F</font> = <font color="Maroon" title="c2">X</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="tops_2.html#T3" title="TOPS_2:th.3">TOPS_2:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">A</font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E4:95"/><i><font color="Green" title="E59">A2</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="tdlat_2.html#V1" title="TDLAT_2:attr.1">domains-family</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:95"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="tdlat_2.html#T65" target="_self" title="TDLAT_2:th.65">Th65</a></i>;<br/>
<a NAME="E5:95"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span></span>)</span> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T68" target="_self" title="TDLAT_2:th.68">Th68</a></i>;<br/>
<a NAME="E6:95"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span></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#V4" title="TOPS_1:attr.4">condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E7:95"/><b>then </b><i><font color="Green" title="E60">A3</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span></span>)</span> <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">T</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D1" title="TDLAT_1:def.1">TDLAT_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">a</font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span></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#K4" title="TDLAT_1:func.4">Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T86" target="_self" title="TDLAT_2:th.86">Th86</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">a</font>
;<br/>

<a NAME="E9:95"/><i><font color="Green" title="E61">A4</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c5">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_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#K4" title="TDLAT_1:func.4">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#D17" title="LATTICE3:def.17">LATTICE3:def 17</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#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="tdlat_2.html#T86" target="_self" title="TDLAT_2:th.86">Th86</a></i>;<br/>
<b>assume </b><a NAME="E2:95_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E3:95_1"/><b>then </b>
<font color="Maroon" title="c7">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T69" target="_self" title="TDLAT_2:th.69">Th69</a></i>;<br/>
<b>hence </b><a NAME="E4:95_1"/>
<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#E7:95"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="tdlat_2.html#T89" target="_self" title="TDLAT_2:th.89">Th89</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:95"/>
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#K4" title="TDLAT_1:func.4">Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span>  st <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c5">a</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_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#K4" title="TDLAT_1:func.4">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#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="tdlat_2.html#T86" target="_self" title="TDLAT_2:th.86">Th86</a></i>;<br/>
<a NAME="E2:95_2"/>
<font color="Maroon" title="c7">B</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">T</font>
 
;<br/>
<a NAME="E3:95_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#V4" title="TOPS_1:attr.4">condensed</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D1" title="TDLAT_1:def.1">TDLAT_1:def 1</a></i>;<br/>
<a NAME="E4:95_2"/><b>then </b><i><font color="Green" title="E62">A5</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#V4" title="TOPS_1:attr.4">condensed</a> )
 
;<br/>
<b>assume </b><a NAME="E5:95_2"/><i><font color="Green" title="E63">A6</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c6">b</font>
 ;<br/>

<a NAME="E6:95_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="c3">F</font> holds <br/><font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_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="E2:95_2_1"/><i><font color="Green" title="E64">A7</font></i>: 
<font color="Maroon" title="c8">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<a NAME="E3:95_2_1"/><b>then </b>
<font color="Maroon" title="c9">C1</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E4:95"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="tdlat_2.html#D2" target="_self" title="TDLAT_2:def.2">Def2</a></i>;<br/>
<a NAME="E4:95_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#V4" title="TOPS_1:attr.4">condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E5:95_2_1"/><b>then </b><i><font color="Green" title="E65">A8</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#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D1" title="TDLAT_1:def.1">TDLAT_1:def 1</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#K4" title="TDLAT_1:func.4">Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T86" target="_self" title="TDLAT_2:th.86">Th86</a></i>;<br/>
<a NAME="E7:95_2_1"/>
<font color="Maroon" title="c10">c</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#E5:95_2"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="tdlat_2.html#E2:95_2_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E8:95_2_1"/>
<font color="Maroon" title="c8">C</font> <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#E5:95_2_1"><i><font color="Green" title="E65">A8</font></i></a>, <a class="ref" href="tdlat_2.html#T89" target="_self" title="TDLAT_2:th.89">Th89</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:95_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font></span>)</span></span>)</span></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#E4:95_2"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="tdlat_2.html#T69" target="_self" title="TDLAT_2:th.69">Th69</a></i>;<br/>
<b>hence </b><a NAME="E8:95_2"/>
<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#E7:95"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="tdlat_2.html#T89" target="_self" title="TDLAT_2:th.89">Th89</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:95"/>
( <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c5">a</font> &amp; ( 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#K4" title="TDLAT_1:func.4">Domains_Lattice</a> <font color="Maroon" title="c1">T</font></span>)</span>  st <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c5">a</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">b</font> ) )
 <b>by </b><i><a class="txt" href="tdlat_2.html#E9:95"><i><font color="Green" title="E61">A4</font></i></a></i>;<br/>


</div>
