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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#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 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="c2">T2</font> implies for <font color="Olive" title="b1">T</font> being   <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> holds <br/> ( <font color="Olive" title="b1">T</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c1">T1</font> &amp; <font color="Olive" title="b1">T</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c2">T2</font> ) )</span><br/>

<b>assume </b><a NAME="E1:90"/><i><font color="Green" title="E41">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="c2">T2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being   <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> holds <br/> ( <font color="Olive" title="b1">T</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c1">T1</font> &amp; <font color="Olive" title="b1">T</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c2">T2</font> )</span><br/>

<b>let </b><font color="Maroon" title="c3">T</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">T</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c1">T1</font> &amp; <font color="Maroon" title="c3">T</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c2">T2</font> )</span><br/>

<i><font color="Green" title="E42">A2</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</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="c2">T2</font>
<b>by </b><i><a class="ref" href="yellow_9.html#D6" target="_self" title="YELLOW_9:def.6">Def6</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>
<b>by </b><i><a class="txt" href="#E1:90"><i><font color="Green" title="E41">A1</font></i></a></i>
;<br/>
<a NAME="E3:90"/>
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="c3">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#D6" target="_self" title="YELLOW_9:def.6">Def6</a></i>;<br/>
<a NAME="E4:90"/><b>then </b>
( 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="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> &amp; 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="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> &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> <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#D5" title="CANTOR_1:def.5">CANTOR_1:def 5</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:90"/><b>then </b>
( 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> &amp; 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> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E6:90"/>
( <font color="Maroon" title="c3">T</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c1">T1</font> &amp; <font color="Maroon" title="c3">T</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c2">T2</font> )
 <b>by </b><i><a class="txt" href="#E1:90"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E2:90"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="yellow_9.html#D5" target="_self" title="YELLOW_9:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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