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

<b>let </b><font color="Maroon">Y0</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<a NAME="E1:1_1_1"/><i><font color="Green">E17</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> )
 
;<br/>
<b>thus </b><a NAME="E2:1_1_1"/>
( <font color="Maroon">Y0</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font> implies ( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; ( for <font color="Olive">G0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> holds <br/> ( <font color="Olive">G0</font> is <a href="pre_topc.html#V3">open</a> iff  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">G0</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> ) ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">Y0</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>
 ;<br/>

<b>hence </b><a NAME="E2:1_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>

<b>thus </b><a NAME="E3:1_1_1_1"/>
for <font color="Olive">G0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> holds <br/> ( <font color="Olive">G0</font> is <a href="pre_topc.html#V3">open</a> iff  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">G0</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G0</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font>;<br/>

<b>thus </b><a NAME="E1:1_1_1_1_1"/>
( <font color="Maroon">G0</font> is <a href="pre_topc.html#V3">open</a> implies  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_1_1_1"/>
<font color="Maroon">G0</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">G0</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E4:1_1_1_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:1_1_1_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E7:1_1_1_1_1_1"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>

<b>thus </b><a NAME="E8:1_1_1_1_1_1"/>
<font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font>
 <b>by </b><i/>;<br/>


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

<b>given </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><a NAME="E2:1_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E3:1_1_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">G</font> = <font color="Maroon">G</font>;<br/><b>thus </b><a NAME="E1:1_1_1_1_1_2"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_2"/>
<font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E5:1_1_1_1_1"/><b>then </b>
<font color="Maroon">G0</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:1_1_1_1_1"/>
<font color="Maroon">G0</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


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


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

<b>assume </b><a NAME="E3:1_1_1"/><i><font color="Green">E23</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; ( for <font color="Olive">G0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> holds <br/> ( <font color="Olive">G0</font> is <a href="pre_topc.html#V3">open</a> iff  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">G0</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font> ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:1_1_1_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E2:1_1_1_2"/>
for <font color="Olive">G0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> holds <br/> ( <font color="Olive">G0</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font> iff  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> &amp; <font color="Olive">G0</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G0</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font>;<br/>

<b>thus </b><a NAME="E1:1_1_1_2_1"/>
( <font color="Maroon">G0</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font> implies  ex <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">G0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> ;<br/>
<b>assume </b><a NAME="E2:1_1_1_2_1_1"/>
<font color="Maroon">G0</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font>
 ;<br/>

<a NAME="E3:1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">M</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E5:1_1_1_2_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:1_1_1_2_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E7:1_1_1_2_1_1"/>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>

<b>thus </b><a NAME="E8:1_1_1_2_1_1"/>
<font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>given </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><a NAME="E2:1_1_1_2_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E3:1_1_1_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y0</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">G0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y0</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">G</font> = <font color="Maroon">G</font>;<br/><b>thus </b><a NAME="E1:1_1_1_2_1_2"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>thus </b><a NAME="E2:1_1_1_2_1_2"/>
<font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:1_1_1_2_1"/><b>then </b>
<font color="Maroon">M</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:1_1_1_2_1"/>
<font color="Maroon">G0</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y0</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:1_1_1"/>
<font color="Maroon">Y0</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>


</div>
