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<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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12">Open_Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="tdlat_1.html#K9">Open_Domains_of</a> <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:111"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K9">Open_Domains_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K9">Open_Domains_of</a> <font color="Maroon">c<sub>1</sub></font> )
 
;<br/>
<a NAME="E3:111"/><b>then </b>
( <font color="Maroon">c<sub>4</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#V6">open_condensed</a> </span>}</span>  &amp; <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#V6">open_condensed</a> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D9">TDLAT_1:def 9</a></i>;<br/>
<a NAME="E4:111"/><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>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V6">open_condensed</a> ) &amp; 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#V6">open_condensed</a> ) )
 
;<br/>
<a NAME="E5:111"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T107">TOPS_1:107</a></i>;<br/>
<b>assume </b><a NAME="E6:111"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E7:111"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:111_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:111_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<a NAME="E3:111_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E6:111"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tdlat_2.html#T104" target="_self">Th104</a></i>;<br/>
<b>hence </b><a NAME="E4:111_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E5:111"><i><font color="Green">E70</font></i></a>, <a class="ref" href="tdlat_2.html#T4" target="_self">Th4</a></i>;<br/>


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

<b>thus </b><a NAME="E8:111"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:111_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:111_2"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E4:111"><i><font color="Green">E69</font></i></a>, <a class="ref" href="tdlat_2.html#T64" target="_self">Th64</a></i>;<br/>
<a NAME="E3:111_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E6:111"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tdlat_2.html#T104" target="_self">Th104</a></i>;<br/>
<b>hence </b><a NAME="E4:111_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>


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


</div>
