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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green" title="E34">A1</font></i>: 
( <font color="Maroon" title="c3">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/>

<a NAME="E2:44"/><i><font color="Green" title="E35">A2</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">PP</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">PP</font> = <font color="Maroon" title="c5">PP</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:44_1"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">PP</font>
 ;<br/><b>take </b><font color="Maroon" title="c8">V</font> = <font color="Maroon" title="c3">V</font>;<br/><b>take </b><font color="Maroon" title="c9">W</font> = <font color="Maroon" title="c4">W</font>;<br/><b>thus </b><a NAME="E2:44_1"/>
( <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c9">W</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c9">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:44"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="borsuk_1.html#E1:44_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:44"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">W</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:44"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="borsuk_1.html#T45" target="_self" title="BORSUK_1:th.45">Th45</a></i>;<br/>


</div>
