<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">T1</font> be    <a href="pre_topc.html#L1" target="_self">TopStruct</a> , <font color="Maroon">T2</font> be    <a href="pre_topc.html#L1" target="_self">TopStruct</a> , <font color="Maroon">S1</font> be    <a href="pre_topc.html#L1" target="_self">TopStruct</a> , <font color="Maroon">S2</font> be    <a href="pre_topc.html#L1" target="_self">TopStruct</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31"/><i><font color="Green">E19</font></i>: 
 <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font>,the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">T1</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font>,the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">T2</font> #)
 <b>and </b><br/><a NAME="E2:31"/><i><font color="Green">E20</font></i>: 
 <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S1</font>,the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">S1</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1" target="_self">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S2</font>,the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">S2</font> #)
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:31"/><i><font color="Green">E21</font></i>: 
 <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">S1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">T1</font>
 <b>and </b><br/><a NAME="E4:31"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font> holds <br/> ( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">S1</font> iff  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</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">T1</font> &amp; <font color="Olive">P</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> <font color="Maroon">S1</font></span>)</span> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D9" target="_self">PRE_TOPC:def 9</a><br/>

<b>thus </b><a NAME="E5:31"/>
 <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">S2</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">T2</font>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D1" target="_self">Def1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D9" target="_self">PRE_TOPC:def 9</a><br/>

<b>thus </b><a NAME="E6:31"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S2</font> holds <br/> ( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">S2</font> iff  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</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">T2</font> &amp; <font color="Olive">P</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> <font color="Maroon">S2</font></span>)</span> ) )
 <b>by </b><i>, , </i>;<br/>


</div>
