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<b>let </b><font color="Maroon">R</font> be    <a href="pre_topc.html#L1">TopStruct</a> , <font color="Maroon">S</font> be    <a href="pre_topc.html#L1">TopStruct</a> , <font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">R</font> is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>
 ;<br/><b>then reconsider </b><font color="Maroon">R1</font> = <font color="Maroon">R</font> as   <a href="pre_topc.html#NM1">TopSpace</a> ;<br/><a NAME="E3:59_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">R1</font> #) is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">R1</font> #) <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#D6">YELLOW_9:def 6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_9.html#D6">YELLOW_9:def 6</a><br/>

<a NAME="E2:59_1_1"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</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/>
<b>hence </b><a NAME="E3:59_1_1"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> is    <a href="cantor_1.html#M2">prebasis</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">R1</font> #)
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:59_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">R</font> #) is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:59"/><i><font color="Green">E35</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">R</font> #) is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">R1</font> = <font color="Maroon">R</font> as   <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i><a class="ref" href="tex_2.html#T12">TEX_2:12</a></i>;<br/>
<a NAME="E4:59"/>
<font color="Maroon">R1</font> is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#D6">YELLOW_9:def 6</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_9.html#D6">YELLOW_9:def 6</a><br/>

<a NAME="E2:59_2"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> is    <a href="cantor_1.html#M2">prebasis</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="pre_topc.html#U1">topology</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/>
<b>hence </b><a NAME="E3:59_2"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> <a href="xboole_0.html#K2">\/</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">R1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:59"/>
<font color="Maroon">R</font> is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>
 ;<br/>


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