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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:86"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="lab"><font color="Green" title="E41">A2</font></span>: 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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D6" target="_self" title="YELLOW_9:def.6">Def6</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E40">A1</font></span></a></span>
;<br/>
<a NAME="E3:86"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
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#NM2" title="CANTOR_1:NM.2">prebasis</a> of <font color="Maroon" title="c3">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D6" target="_self" title="YELLOW_9:def.6">Def6</a></span>;<br/>
<a NAME="E4:86"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E5:86"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E6:86"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="tops_2.html#T78" title="TOPS_2:th.78">TOPS_2:78</a></span>;<br/>
<a NAME="E7:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A7</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E8:86"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:86"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E6:86"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:86"/>
( <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E7:86"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="yellow_9.html#D5" target="_self" title="YELLOW_9:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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