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<b>let </b><font color="Maroon">T</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">S</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">R</font> be    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><a NAME="E1:30"/>
( <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> or <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> )
 ;<br/>

<a NAME="E2:30"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="xboole_0.html#K2">\/</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:30"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="xboole_0.html#K2">\/</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#D6">YELLOW_9:def 6</a></i>;<br/>
<a NAME="E4:30"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="xboole_0.html#K2">\/</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:30"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">R</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>


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