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<div class="add">

<b>let </b><font color="Maroon">T1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T2</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">B1</font> be    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T1</font>;<br/><b>let </b><font color="Maroon">B2</font> be    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T2</font>;<br/><b>let </b><font color="Maroon">T</font> be    <a href="yellow_9.html#M3" target="_self">Refinement</a> of <font color="Maroon">T1</font>,<font color="Maroon">T2</font>;<br/>







<b>set </b><font color="Maroon">BB</font> = <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E2:92"/><i><font color="Green">E32</font></i>: 
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:92"/><i><font color="Green">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:92"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> &amp; <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a>, <a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>
<a NAME="E5:92"/>
( the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:92"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="cantor_1.html#T4">CANTOR_1:4</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:92"/><b>then </b>
( <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E8:92"/><b>then </b>
(  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="cantor_1.html#T15">CANTOR_1:15</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:92"/><b>then </b><i><font color="Green">E38</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font> &amp;  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E10:92"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:92"/>
<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font></span>)</span> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">BB</font> = <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font></span>)</span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:92_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E2:92_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:92_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:92_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">Y</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self">Th3</a>, <a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">Y1</font> = <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T1</font> ;<br/><b>reconsider </b><font color="Maroon">Y2</font> = <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T2</font> ;<br/><i><font color="Green">E56</font></i>: <font color="Maroon">Y</font> = 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="yellow_9.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">Y1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y2</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/><a NAME="E10:92_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">cT1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> ;<br/><a NAME="E12:92_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">cT2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> ;<br/><a NAME="E14:92_1"/>
( <font color="Maroon">cT1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> &amp; <font color="Maroon">cT2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/><a NAME="E15:92_1"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">cT1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">cT2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">cT1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">cT2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>thus </b><a NAME="E16:92_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:92_1_2_1"/>
( ( <font color="Maroon">Y1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cT1</font> ) or ( <font color="Maroon">Y1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cT2</font> ) or ( <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">Y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">Y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_1_1"><b>suppose </b></a><a NAME="E1:92_1_2_1_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">Y1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cT1</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">b1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E3:92_1_2_1_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">cT1</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E10:92"><i><font color="Green">E52</font></i></a>, </i>;<br/><a NAME="E4:92_1_2_1_1"/>
( <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T4" target="_self">Th4</a>, , <a class="txt" href="yellow_9.html#E1:92_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="yellow_9.html#E4:92_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:92_1_2_1_1"/><b>then </b>
( <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">b1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">b1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:92_1_2_1_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E4:92_1"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_1_2"><b>suppose </b></a><a NAME="E1:92_1_2_1_2"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">Y1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">cT2</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">b2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font><b> such that </b><br/><a NAME="E3:92_1_2_1_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">b2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">cT2</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E10:92"><i><font color="Green">E52</font></i></a>, </i>;<br/><a NAME="E4:92_1_2_1_2"/>
( <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T4" target="_self">Th4</a>, , <a class="txt" href="yellow_9.html#E1:92_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="yellow_9.html#E4:92_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:92_1_2_1_2"/><b>then </b>
( <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">b2</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">b2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:92_1_2_1_2"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E4:92_1"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_1_3"><b>suppose </b></a><a NAME="E1:92_1_2_1_3"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E95</font></i>: <font color="Maroon">a</font> = 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Y2</font>
<b>by </b><i><a class="ref" href="yellow_9.html#T4" target="_self">Th4</a>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y2</font>
<b>by </b><i><a class="txt" href="yellow_9.html#E1:92_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
;<br/><a NAME="E3:92_1_2_1_3"/>
<font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E4:92_1_2_1_3"/><b>then </b>
( <font color="Maroon">Y2</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T4" target="_self">Th4</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E5:92_1_2_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E4:92_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/><a NAME="E6:92_1_2_1_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">B2</font> )
 <b>by </b><i>, <a class="ref" href="cantor_1.html#T5">CANTOR_1:5</a></i>;<br/><b>then consider </b><font color="Maroon">Z</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T2</font><b> such that </b><br/><a NAME="E8:92_1_2_1_3"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:92_1_2_1_3"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self">Th3</a>, <a class="txt" href="yellow_9.html#E6:92_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E11:92_1_2_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E6:92_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yellow_9.html#E9:92_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E12:92_1_2_1_3"/><b>then </b><i><font color="Green">E99</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E6:92_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yellow_9.html#E9:92_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E13:92_1_2_1_3"/><b>then </b>
( <font color="Maroon">z</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E14:92_1_2_1_3"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E9:92_1"><i><font color="Green">E56</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_1_4"><b>suppose </b></a><a NAME="E1:92_1_2_1_4"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E101</font></i>: <font color="Maroon">a</font> = 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Y1</font>
<b>by </b><i><a class="ref" href="yellow_9.html#T4" target="_self">Th4</a>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y1</font>
<b>by </b><i><a class="txt" href="yellow_9.html#E1:92_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
;<br/><a NAME="E3:92_1_2_1_4"/>
<font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E4:92_1_2_1_4"/><b>then </b>
( <font color="Maroon">Y1</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T4" target="_self">Th4</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E5:92_1_2_1_4"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E4:92_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/><a NAME="E6:92_1_2_1_4"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">B1</font> )
 <b>by </b><i>, <a class="ref" href="cantor_1.html#T5">CANTOR_1:5</a></i>;<br/><b>then consider </b><font color="Maroon">Z</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E8:92_1_2_1_4"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:92_1_2_1_4"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self">Th3</a>, <a class="txt" href="yellow_9.html#E6:92_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E11:92_1_2_1_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font>
 <b>by </b><i><a class="txt" href="yellow_9.html#E6:92_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yellow_9.html#E9:92_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E12:92_1_2_1_4"/><b>then </b><i><font color="Green">E148</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E6:92_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yellow_9.html#E9:92_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E13:92_1_2_1_4"/><b>then </b>
( <font color="Maroon">z</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E14:92_1_2_1_4"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E9:92_1"><i><font color="Green">E56</font></i></a>, </i>;<br/><b>thus </b><a NAME="E15:92_1_2_1_4"/>
verum
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="92_1_2_1_5"><b>suppose </b></a><a NAME="E1:92_1_2_1_5"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>then </b><i><font color="Green">E150</font></i>: <font color="Maroon">a</font> = 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Y</font>
<b>by </b><i><a class="ref" href="yellow_9.html#T4" target="_self">Th4</a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">Y1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">Y2</font></span>)</span>
<b>by </b><i>, <a class="txt" href="yellow_9.html#E1:92_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">Y1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y2</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_9.html#E1:92_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y2</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_9.html#E1:92_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
;<br/><a NAME="E3:92_1_2_1_5"/>
( <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E4:92_1_2_1_5"/><b>then </b>
( <font color="Maroon">Y1</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">Y2</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> &amp; <font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T4" target="_self">Th4</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E5:92_1_2_1_5"/><b>then </b>
(  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/><a NAME="E6:92_1_2_1_5"/><b>then </b><i><font color="Green">E177</font></i>: 
(  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">B1</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">B2</font> )
 <b>by </b><i>, <a class="ref" href="cantor_1.html#T5">CANTOR_1:5</a></i>;<br/><b>then consider </b><font color="Maroon">Z1</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T1</font><b> such that </b><br/><a NAME="E8:92_1_2_1_5"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">Z1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Z1</font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/><a NAME="E9:92_1_2_1_5"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self">Th3</a>, <a class="txt" href="yellow_9.html#E4:92_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">z1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:92_1_2_1_5"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z1</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E9:92_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">Z2</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T2</font><b> such that </b><br/><a NAME="E13:92_1_2_1_5"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">Z2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font> &amp;  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Z2</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E6:92_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/><a NAME="E14:92_1_2_1_5"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y2</font>
 <b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self">Th3</a>, <a class="txt" href="yellow_9.html#E4:92_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">z2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:92_1_2_1_5"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z2</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z2</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E6"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E17:92_1_2_1_5"/>
( <font color="Maroon">z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font> &amp; <font color="Maroon">z1</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Z1</font> &amp; <font color="Maroon">z2</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Z2</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E9:92_1"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="yellow_9.html#E6"><i><font color="Green">Lemma58</font></i></a>, , <a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E18:92_1_2_1_5"/><b>then </b><i><font color="Green">E184</font></i>: 
( <font color="Maroon">z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="yellow_9.html#E4:92_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="yellow_9.html#E9:92_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="yellow_9.html#E6"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E19:92_1_2_1_5"/><b>then </b>
( <font color="Maroon">z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z2</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z2</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T3" target="_self">Th3</a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E20:92_1_2_1_5"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:92"/>
<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font></span>)</span> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/>


</div>
