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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">BB</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>



<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1"/>
<font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><a NAME="E2:41_1"/><b>then </b>
( the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>hence </b><a NAME="E3:41_1"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T1">ZFMISC_1:1</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E2:41"/>
( <font color="Maroon">BB</font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font> implies  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">BB</font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font>
 ;<br/>

<a NAME="E2:41_2"/><b>then </b>
<font color="Maroon">BB</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#D5">CANTOR_1:def 5</a></i>;<br/>
<a NAME="E3:41_2"/><b>then </b>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</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#T16">CANTOR_1:16</a></i>;<br/>
<a NAME="E4:41_2"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</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#T5">CANTOR_1:5</a></i>;<br/>
<b>consider </b><font color="Maroon">A</font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:41_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font>
 <b>by </b><i>, <a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>
<a NAME="E7:41_2"/>
( the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">A</font> &amp;  <a href="cantor_1.html#K1">UniCl</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">BB</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="cantor_1.html#T9">CANTOR_1:9</a>, <a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<a NAME="E8:41_2"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</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">BB</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E9:41_2"/>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>


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

<b>assume </b><a NAME="E3:41"/><i><font color="Green">E38</font></i>: 
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 ;<br/>

<a NAME="E4:41"/><b>then </b>
( <font color="Maroon">BB</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font> &amp;  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</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#T4">CANTOR_1:4</a>, <a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<a NAME="E5:41"/><b>then </b>
<font color="Maroon">BB</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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:41"/>
<font color="Maroon">BB</font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>


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