<?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#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K8">Closed_Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<a NAME="E1:103"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K8">Closed_Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<a NAME="E2:103"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T94" target="_self">Th94</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tops_2.html#T3">TOPS_2:3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E4:103"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tdlat_2.html#V2" target="_self">closed-domains-family</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E2:103"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tdlat_2.html#T72" target="_self">Th72</a></i>;<br/>
<a NAME="E5:103"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="tops_1.html#V5">closed_condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T76" target="_self">Th76</a></i>;<br/>
<a NAME="E6:103"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font></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#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E7:103"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D5">TDLAT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K8">Closed_Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T94" target="_self">Th94</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<a NAME="E9:103"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_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#K8">Closed_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#D17">LATTICE3:def 17</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#K5">Closed_Domains_of</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tdlat_2.html#T94" target="_self">Th94</a></i>;<br/>
<b>assume </b><a NAME="E2:103_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:103_1"/><b>then </b>
<font color="Maroon">c<sub>7</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="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T77" target="_self">Th77</a></i>;<br/>
<b>hence </b><a NAME="E4:103_1"/>
<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#E7:103"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tdlat_2.html#T97" target="_self">Th97</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:103"/>
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#K8">Closed_Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R1">[=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_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#K8">Closed_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#K5">Closed_Domains_of</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tdlat_2.html#T94" target="_self">Th94</a></i>;<br/>
<a NAME="E2:103_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E3:103_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#V5">closed_condensed</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D5">TDLAT_1:def 5</a></i>;<br/>
<a NAME="E4:103_2"/><b>then </b><i><font color="Green">E69</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#V5">closed_condensed</a> )
 
;<br/>
<b>assume </b><a NAME="E5:103_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E6:103_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>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_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="E2:103_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:103_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> is <a href="tops_1.html#V5">closed_condensed</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E4:103"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tdlat_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:103_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#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E5:103_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D5">TDLAT_1:def 5</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#K8">Closed_Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="tdlat_2.html#T94" target="_self">Th94</a></i>;<br/>
<a NAME="E7:103_2_1"/>
<font color="Maroon">c<sub>10</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#E5:103_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="tdlat_2.html#E2:103_2_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<b>hence </b><a NAME="E8:103_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <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#E5:103_2_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="tdlat_2.html#T97" target="_self">Th97</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:103_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>3</sub></font></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#E4:103_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="tdlat_2.html#T77" target="_self">Th77</a></i>;<br/>
<b>hence </b><a NAME="E8:103_2"/>
<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#E7:103"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tdlat_2.html#T97" target="_self">Th97</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:103"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( 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#K8">Closed_Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4">is_less_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R1">[=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="tdlat_2.html#E9:103"><i><font color="Green">E68</font></i></a></i>;<br/>


</div>
