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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</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="9_1_1"><b>suppose </b></a><a NAME="E1:9_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">T</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #);<br/><a NAME="E2:9_1_1"/><i><font color="Green">E28</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #) <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#T8">CANTOR_1:8</a></i>;<br/><a NAME="E3:9_1_1"/>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="cantor_1.html#T1">CANTOR_1:1</a></i>;<br/><b>hence </b><a NAME="E4:9_1_1"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #) <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #)
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a><br/><b>hence </b><a NAME="E5:9_1_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #)  st <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #) holds <br/> <a href="setfam_1.html#K5">union</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #)
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>thus </b><a NAME="E6:9_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #)  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #) &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #) holds <br/><font color="Olive">a</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #)
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2"><b>suppose </b></a><a NAME="E1:9_1_2"/>
<font color="Maroon">X</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="E2:9_1_2"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #) is <a href="pre_topc.html#V2">TopSpace-like</a>
 <b>by </b><i><a class="ref" href="cantor_1.html#T17">CANTOR_1:17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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