<?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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E1:106"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> implies the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:106_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:106_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:106_1_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:106_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">S</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">P</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">S</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">P</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="Olive">S</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:106_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E5:106_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">H</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">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> )
 ;<br/><b>reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">H</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><a NAME="E7:106_1_1_1"/>
<font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</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#D1">PRE_TOPC:def 1</a></i>;<br/><a NAME="E8:106_1_1_1"/><b>then </b>
( <font color="Maroon">H1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</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/><a NAME="E9:106_1_1_1"/><b>then </b>
<font color="Maroon">H1</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T37">TOPS_1:37</a></i>;<br/><b>hence </b><a NAME="E10:106_1_1_1"/>
<font color="Maroon">W</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="tmap_1.html#T15" target="_self">Th15</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:106_1_1"/>
<font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:106_1"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:106_1"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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

<b>thus </b><a NAME="E2:106"/>
( the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="tmap_1.html#K5" target="_self">-extension_of_the_topology_of</a> <font color="Maroon">X</font> implies <font color="Maroon">A</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="tmap_1.html#T13" target="_self">Th13</a></i>;<br/>


</div>
