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<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>assume </b><a NAME="E1:93"/><i><font color="Green">E32</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</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">B1</font> = the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font>;<br/>
<b>set </b><font color="Maroon">B2</font> = the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font>;<br/>
<a NAME="E2:93"/>
 <a href="cantor_1.html#K1">UniCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T6">CANTOR_1:6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B1</font> = the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font> as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T1</font> <b>by </b><i/>;<br/>
<a NAME="E4:93"/>
 <a href="cantor_1.html#K1">UniCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> <a href="hidden.html#R1">=</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#T6">CANTOR_1:6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B2</font> = the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T2</font> <b>by </b><i/>;<br/>
<a NAME="E6:93"/><i><font color="Green">E33</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> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:93"/><i><font color="Green">E35</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E8:93"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:93_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 ;<br/>

<a NAME="E2:93_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</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">a</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> )
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:93_1"/>
<font color="Maroon">a</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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:93"/>
<font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:93_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 ;<br/>

<a NAME="E2:93_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</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">a</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> )
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:93_2"/>
<font color="Maroon">a</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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:93"/><b>then </b>
<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="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E11:93"/>
 <a href="setfam_1.html#K3">INTERSECTION</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T2</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>


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