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

<b>let </b><font color="Maroon">T</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">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14">space</a> <font color="Maroon">S</font></span>)</span>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>







<b>assume </b><a NAME="E1:4"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">A</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>;<br/>
<i><font color="Green">E21</font></i>: <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="eqrel_1.html#D6">EQREL_1:def 6</a></i>
<br/>.= 
 <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K14">space</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>
;<br/>
<b>thus </b><a NAME="E4:4"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> implies <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_2"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:4_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14">space</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</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#D6">PRE_TOPC:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">om</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14">space</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>;<br/>
<a NAME="E4:4_2"/>
<font color="Maroon">om</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14">space</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, , <a class="ref" href="borsuk_1.html#T69">BORSUK_1:69</a></i>;<br/>
<a NAME="E6:4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</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>hence </b><a NAME="E7:4_2"/>
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>


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

<b>thus </b><a NAME="E5:4"/>
( <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a> implies <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_3"/>
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>
<a NAME="E3:4_3"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">om</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14">space</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="borsuk_1.html#D10">BORSUK_1:def 10</a></i>;<br/>
<a NAME="E5:4_3"/>
<font color="Maroon">om</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14">space</a> <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="borsuk_1.html#T69">BORSUK_1:69</a></i>;<br/>
<a NAME="E6:4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K14">space</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</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/>
<b>hence </b><a NAME="E7:4_3"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>


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


</div>
