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



<b>assume </b><a NAME="E1:33"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font> &amp; <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> )
 ;<br/>

<a NAME="E2:33"/><b>then </b>
<font color="Maroon">K</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>then reconsider </b><font color="Maroon">K'</font> = <font color="Maroon">K</font>, <font color="Maroon">O'</font> = <font color="Maroon">O</font> as   <a href="struct_0.html#NM3">Subset-Family</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>reconsider </b><font color="Maroon">K'</font> = <font color="Maroon">K'</font>, <font color="Maroon">O'</font> = <font color="Maroon">O'</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">K'</font> = <font color="Maroon">K'</font>, <font color="Maroon">O'</font> = <font color="Maroon">O'</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E6:33"/><i><font color="Green">E34</font></i>: 
(  <a href="cantor_1.html#K2">FinMeetCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp;  <a href="cantor_1.html#K1">UniCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</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>, <a class="ref" href="cantor_1.html#T6">CANTOR_1:6</a></i>;<br/>
<a NAME="E7:33"/><b>then </b><i><font color="Green">E36</font></i>: 
(  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">O'</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp;  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">K'</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">O'</font> )
 
<b>by </b><i>, <a class="ref" href="cantor_1.html#T9">CANTOR_1:9</a></i>;<br/>
<a NAME="E8:33"/>
(  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">O'</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp;  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">K'</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">O'</font> )
 
<b>by </b><i>, , <a class="ref" href="cantor_1.html#T16">CANTOR_1:16</a></i>;<br/>
<a NAME="E9:33"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <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">O'</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> &amp;  <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">K'</font></span>)</span> <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">O'</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="cantor_1.html#T9">CANTOR_1:9</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon">K</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 ;<br/><a NAME="E2:33_1"/><b>then </b>
 <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">K'</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_9.html#T22">YELLOW_9:22</a></i>;<br/><a NAME="E3:33_1"/><b>then </b>
 <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">O'</font> <a href="hidden.html#R1">=</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_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E4:33_1"/>
<font color="Maroon">O</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_9.html#T22">YELLOW_9:22</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E11:33"/>
<font color="Maroon">K</font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font>
 ;<br/>

<a NAME="E12:33"/><b>then </b>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">K'</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T23">YELLOW_9:23</a></i>;<br/>
<a NAME="E13:33"/><b>then </b>
 <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">K'</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T22">YELLOW_9:22</a></i>;<br/>
<a NAME="E14:33"/><b>then </b>
 <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">O'</font></span>)</span> <a href="hidden.html#R1">=</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_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E15:33"/><b>then </b>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">O'</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T22">YELLOW_9:22</a></i>;<br/>
<b>hence </b><a NAME="E16:33"/>
<font color="Maroon">O</font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_9.html#T23">YELLOW_9:23</a></i>;<br/>


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