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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> );<br/>
<b>consider </b><font color="Maroon">top1</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">P</font><b> such that </b><br/><a NAME="E2:23_1_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">Z</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">P</font> holds <br/> ( <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">top1</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Z</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">top1</font> = <font color="Maroon">top1</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">P</font> ;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">P</font>,<font color="Maroon">top1</font> #);<br/>
<a NAME="E4:23_1_1"/><i><font color="Green">E20</font></i>: 
 <a href="pre_topc.html#K2" target="_self">[#]</a> <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">P</font>,<font color="Maroon">top1</font> #) <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">T</font>
 
;<br/>
<a NAME="E5:23_1_1"/>
for <font color="Olive">Z</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">P</font>,<font color="Maroon">top1</font> #) holds <br/> ( <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">top1</font> iff  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">T</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">P</font>,<font color="Maroon">top1</font> #)</span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">P</font>,<font color="Maroon">top1</font> #);<br/>

<b>thus </b><a NAME="E1:23_1_1_1"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">top1</font> implies  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">P</font>,<font color="Maroon">top1</font> #)</span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">top1</font>
 ;<br/>

<a NAME="E2:23_1_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Z</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:23_1_1_1_1"/>
 ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">P</font>,<font color="Maroon">top1</font> #)</span>)</span> )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:23_1_1_1"/>
(  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">P</font>,<font color="Maroon">top1</font> #)</span>)</span> ) implies <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">top1</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> =  <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# <font color="Maroon">P</font>,<font color="Maroon">top1</font> #) as  <a href="pre_topc.html#V1" target="_self">strict</a>  <a href="pre_topc.html#M1" target="_self">SubSpace</a> of <font color="Maroon">T</font> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E7:23_1_1"/>
 <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 ;<br/>


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