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<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>



<a NAME="E1:31"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">F</font> holds <br/> <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A0</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:31_1"/>
<font color="Maroon">A0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> :  ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:31_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A0</font>
 <b>and </b><br/><a NAME="E6:31_1"/><i><font color="Green">E23</font></i>: 
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>
<b>consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:31_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E9:31_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E6:31_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E10:31_1"/>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<b>hence </b><a NAME="E11:31_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E5:31_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tdlat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1"/>
(  <a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1"><b>suppose </b></a><a NAME="E1:31_2_1_1"/>
 <a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:31_2_1_1"/><b>then </b>
(  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> &amp;  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> &amp;  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="tdlat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="setfam_1.html#T2">SETFAM_1:2</a></i>;<br/><b>hence </b><a NAME="E3:31_2_1_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">F</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2"><b>suppose </b></a><a NAME="E1:31_2_1_2"/>
 <a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_2_1_2"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tdlat_2.html#T4" target="_self">Th4</a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" target="_self">Int</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>


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