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

<b>let </b><font color="Maroon">X</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">X0</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>



<b>reconsider </b><font color="Maroon">S</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X0</font> #) as    <a href="pre_topc.html#L1">TopStruct</a>  ;<br/>
<a NAME="E2:13"/>
<font color="Maroon">S</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_1"/><i><font color="Green">E23</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X0</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> )
 
;<br/>
<b>hence </b><a NAME="E2:13_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a><br/>

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<b>thus </b><a NAME="E3:13_1"/>
( <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> implies  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">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">[#]</a> <font color="Maroon">S</font></span>)</span> ) )
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>

<b>given </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><a NAME="E4:13_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span> )
 ;<br/>

<b>thus </b><a NAME="E5:13_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X0</font> #) is  <a href="pre_topc.html#V1">strict</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>
 ;<br/>


</div>
