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<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/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1"/>
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">F</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_1_1"/>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:40_1_1_1"/><b>then </b>
 <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="tdlat_2.html#T9" target="_self" title="TDLAT_2:th.9">Th9</a></i>;<br/><a NAME="E3:40_1_1_1"/><b>then </b>
 <a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="tdlat_2.html#T19" target="_self" title="TDLAT_2:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E4:40_1_1_1"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p3">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tdlat_2.html#T9" target="_self" title="TDLAT_2:th.9">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_2"/>
<font color="Maroon" title="c2">F</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/><div class="add"><a NAME="E2:40_1_1_2"/><b>then </b>
 <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</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> 
 <b>by </b><i><a class="ref" href="tdlat_2.html#T9" target="_self" title="TDLAT_2:th.9">Th9</a></i>;<br/><a NAME="E3:40_1_1_2"/><b>then </b>
 <a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="tdlat_2.html#T19" target="_self" title="TDLAT_2:th.19">Th19</a></i>;<br/><a NAME="E4:40_1_1_2"/><b>then </b><i><font color="Green" title="E24">A1</font></i>: 
 <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="tdlat_2.html#T9" target="_self" title="TDLAT_2:th.9">Th9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:40_1_1_2_1"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:40_1_1_2_1"/>
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:40_1_1_2_1_1"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">A</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>consider </b><font color="Maroon" title="c6">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:40_1_1_2_1_1"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">B</font>
 <b>and </b><br/><a NAME="E5:40_1_1_2_1_1"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E1:40_1_1_2_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="pcomps_1.html#D3" title="PCOMPS_1:def.3">PCOMPS_1:def 3</a></i>;<br/>
<a NAME="E6:40_1_1_2_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E1:40_1_1_2_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="tdlat_2.html#E5:40_1_1_2_1_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:40_1_1_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:40_1_1_2_1_1"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:40_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p3">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tdlat_2.html#E4:40_1_1_2"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:40_1_1_2"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p3">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:40"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <span class="p2">(<span class="default"><a href="tdlat_2.html#K1" title="TDLAT_2:func.1">Int</a> <span class="p3">(<span class="default"><a href="tdlat_2.html#NK1" title="TDLAT_2:NK.1">Cl</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span></span>)</span>
 ;<br/>


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