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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X1</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X1</font>;<br/>







<b>assume </b><a NAME="E1:12"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E2:12"/>
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X1</font></span>)</span> holds <br/><font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X1</font></span>)</span>
 ;<br/>

<a NAME="E2:12_1"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </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><br/><a NAME="E4:12_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:12_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:12_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/>
<a NAME="E7:12_1"/>
 ex <font color="Olive">Q1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X1</font> st <br/>( <font color="Olive">Q1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">Q1</font> = <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">X1</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X1</font> ;<br/>
<b>take </b>
<font color="Maroon">Q1</font>
;<br/>

<a NAME="E2:12_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">Q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E3:12_1_1"/>
<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>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:12_1_1"/><b>then </b>
<font color="Maroon">Q1</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</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="E5:12_1_1"/>
( <font color="Maroon">Q1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Q1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font> )
 <b>by </b><i>, <a class="ref" href="connsp_2.html#D1" target="_self">Def1</a>, , <a class="ref" href="connsp_2.html#D2" target="_self">Def2</a>, , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:12_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12"/>
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div>
