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<b>let </b><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:36"/><i><font color="Green" title="E27">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font>
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green" title="E28">A2</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T</font>
 <b>and </b><br/><a NAME="E3:36"/><i><font color="Green" title="E29">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font>
 <b>and </b><br/><a NAME="E4:36"/><i><font color="Green" title="E30">A4</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_9.html#D5" title="YELLOW_9:def.5">YELLOW_9:def 5</a><br/>

<b>let </b><font color="Maroon" title="c4">R</font> be    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font>;<br/>

<a NAME="E5:36"/><i><font color="Green" title="E31">A5</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font> &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font> is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c4">R</font> )
 
<b>by </b><i><a class="txt" href="waybel33.html#E1:36"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="waybel33.html#E3:36"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="yellow_9.html#D6" title="YELLOW_9:def.6">YELLOW_9:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:36"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_9.html#D5" title="YELLOW_9:def.5">YELLOW_9:def 5</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">S</font> = the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font> as    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c4">R</font> <b>by </b><i><a class="ref" href="yellow_9.html#D6" title="YELLOW_9:def.6">YELLOW_9:def 6</a></i>;<br/>
<a NAME="E8:36"/>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">S</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c4">R</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T23" title="YELLOW_9:th.23">YELLOW_9:23</a></i>;<br/>
<a NAME="E9:36"/><b>then </b><i><font color="Green" title="E32">A6</font></i>: 
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">R</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T22" title="YELLOW_9:th.22">YELLOW_9:22</a></i>;<br/>
<a NAME="E10:36"/>
<font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="txt" href="waybel33.html#E2:36"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="waybel33.html#E4:36"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E11:36"/><b>then </b>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="txt" href="waybel33.html#E5:36"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="cantor_1.html#T16" title="CANTOR_1:th.16">CANTOR_1:16</a></i>;<br/>
<a NAME="E12:36"/><b>then </b>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel33.html#E5:36"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="waybel33.html#E9:36"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="cantor_1.html#T9" title="CANTOR_1:th.9">CANTOR_1:9</a></i>;<br/>
<b>hence </b><a NAME="E13:36"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#T7" title="CANTOR_1:th.7">CANTOR_1:7</a></i>;<br/>


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