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

<b>let </b><font color="Maroon">X</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">Y</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">XV</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>



<b>set </b><font color="Maroon">S</font> = <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>
<b>set </b><font color="Maroon">T</font> = <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>
<a NAME="E1:29"/><i><font color="Green">E19</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XV</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E2:29"/><i><font color="Green">E26</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E3:29"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XV</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green">E40</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E5:29"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i>, <a class="txt" href="borsuk_3.html#E1:29"><i><font color="Green">E19</font></i></a>, , <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E6:29"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/> ( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> iff  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:29_1_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/><a NAME="E2:29_1_1"/><b>then </b>
<font color="Maroon">P'</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">A</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E4:29_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">P'</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">X1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> ) ) )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/><b>set </b><font color="Maroon">AA</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y2</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">Y2</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> st <br/>( <font color="Olive">Y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y2</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> ;<br/><a NAME="E5:29_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y2</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">Y2</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> st <br/>( <font color="Olive">Y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y2</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y2</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">Y2</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> st <br/>( <font color="Olive">Y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y2</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Xx1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">Yy2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:29_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Xx1</font>,<font color="Maroon">Yy2</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp;  ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> st <br/>( <font color="Olive">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Yy2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span> &amp; <font color="Maroon">Xx1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Yy2</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Xx1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:29_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">AA</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y2</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">Y2</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> st <br/>( <font color="Olive">Y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">Y2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y2</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/><b>reconsider </b><font color="Maroon">AA</font> = <font color="Maroon">AA</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/><a NAME="E8:29_1_1"/>
<font color="Maroon">AA</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Xx1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">Yy2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:29_1_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Xx1</font>,<font color="Maroon">Yy2</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp;  ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> st <br/>( <font color="Olive">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Yy2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span> &amp; <font color="Maroon">Xx1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Yy2</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Xx1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 ;<br/>
<b>set </b><font color="Maroon">A'</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E4:29_1_1_2"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:29_1_1_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A'</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">t</font></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<a NAME="E6:29_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">A'</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>
<a NAME="E7:29_1_1_2"/>
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">Y1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">X1</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">Y1</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A'</font>
 ;<br/>

<a NAME="E2:29_1_1_2_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Xx1</font>,<font color="Maroon">Yy2</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:29_1_1_2_2"/>
( <font color="Maroon">Xx1</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">Yy2</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/>
<b>hence </b><a NAME="E4:29_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">Y1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">X1</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">Y1</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> 
 <b>by </b><i><a class="txt" href="borsuk_3.html#E2:29"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:29_1_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Olive">As</font></span>)</span> where <font color="Olive">As</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> : <font color="Olive">As</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">X1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">Y1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">X1</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">Y1</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>  </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:29_1_1_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:29_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/><a NAME="E10:29_1_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1_1_3"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_3_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_3_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A1</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i/>;<br/>
<a NAME="E5:29_1_1_3_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:29_1_1_3_1"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>consider </b><font color="Maroon">X2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">Y2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font><b> such that </b><br/><a NAME="E8:29_1_1_3_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">X2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">p1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">p2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:29_1_1_3_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E11:29_1_1_3_1"/>
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">XV</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">Q1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E13:29_1_1_3_1"/><i><font color="Green">E104</font></i>: 
( <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">X</font> &amp; <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</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">Q1</font> = <font color="Maroon">Q1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<b>set </b><font color="Maroon">EX</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Q1</font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/>
<a NAME="E15:29_1_1_3_1"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q1</font>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E16:29_1_1_3_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Q1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E17:29_1_1_3_1"/>
( <font color="Maroon">Q1</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Y2</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E18:29_1_1_3_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X2</font>,<font color="Maroon">Q1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">Xx1</font>,<font color="Olive">Yy2</font></span><a href="borsuk_1.html#K6">:]</a></span> where <font color="Olive">Xx1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Olive">Yy2</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  ex <font color="Olive">Z1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> st <br/>( <font color="Olive">Z1</font> <a href="hidden.html#R1">=</a> <font color="Olive">Yy2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span> &amp; <font color="Olive">Xx1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Yy2</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">Xx1</font>,<font color="Olive">Z1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E19:29_1_1_3_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>hence </b><a NAME="E20:29_1_1_3_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E2:29"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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

<b>thus </b><a NAME="E2:29_1_1_3"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">h</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_3_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:29_1_1_3_2"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> )
 
<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">A2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:29_1_1_3_2"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">Xx1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">Yy2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:29_1_1_3_2"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Xx1</font>,<font color="Maroon">Yy2</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp;  ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> st <br/>( <font color="Olive">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Yy2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span> &amp; <font color="Maroon">Xx1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Yy2</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Xx1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E2:29"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">p1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">p2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:29_1_1_3_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xx1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Yy2</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E2:29"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">Yy1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font><b> such that </b><br/><a NAME="E10:29_1_1_3_2"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">Yy1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Yy2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span> &amp; <font color="Maroon">Xx1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Yy2</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Xx1</font>,<font color="Maroon">Yy1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<a NAME="E11:29_1_1_3_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">XV</font>
 
<b>by </b><i>, , <a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:29_1_1_3_2"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E13:29_1_1_3_2"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Yy2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:29_1_1_3_2"/><b>then </b>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">Xx1</font>,<font color="Maroon">Yy1</font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E15:29_1_1_3_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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


</div><b>end;</b></div><b>hence </b><a NAME="E11:29_1_1"/>
 ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> st <br/>( <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><a NAME="E3:29_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <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> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">Q'</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<a NAME="E5:29_1"/>
<font color="Maroon">Q'</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">A</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E7:29_1"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">Q'</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">X1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> ) ) )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<b>reconsider </b><font color="Maroon">oS</font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i>, <a class="txt" href="borsuk_3.html#E1:29"><i><font color="Green">E19</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">AA</font> = <font color="Maroon">A</font> <a href="tops_2.html#K1">|</a> <font color="Maroon">oS</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">oS</font></span>)</span> ;<br/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">oS</font></span>)</span> = 
<font color="Maroon">oS</font>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">AA</font> = <font color="Maroon">AA</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon">AA</font> = <font color="Maroon">AA</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<a NAME="E14:29_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E15:29_1"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">oS</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E16:29_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tops_2.html#T44">TOPS_2:44</a></i>;<br/>
<a NAME="E17:29_1"/><b>then </b><i><font color="Green">E120</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">oS</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E18:29_1"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">oS</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_3"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">oS</font>
 ;<br/>

<a NAME="E2:29_1_3"/><b>then </b><i><font color="Green">E122</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">A</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">oS</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">A1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:29_1_3"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:29_1_3"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">oS</font>
 
<b>by </b><i>, <a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A1</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E7:29_1_3"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">oS</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="tops_2.html#T41">TOPS_2:41</a></i>;<br/>
<b>hence </b><a NAME="E8:29_1_3"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:29_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">AA</font>
 
<b>by </b><i>, , <a class="txt" href="borsuk_3.html#E2:29"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E20:29_1"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> holds <br/> ex <font color="Olive">X1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:29_1_4"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e'</font> = <font color="Maroon">e</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">oS</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">R</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E4:29_1_4"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">R</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">oS</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font> )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="tops_2.html#D3">TOPS_2:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">X1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>, <font color="Maroon">Y1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:29_1_4"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">X1</font>,<font color="Maroon">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, <a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E7:29_1_4"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:29_1_4"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">e'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">Y1</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a>, , <a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D1</font> = <font color="Maroon">X1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">Y</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon">D2</font> = <font color="Maroon">Y1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">XV</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> ;<br/>
<a NAME="E11:29_1_4"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">D1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E12:29_1_4"/>
<font color="Maroon">Y1</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="E13:29_1_4"/><b>then </b>
<font color="Maroon">D2</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">XV</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E14:29_1_4"/><b>then </b>
<font color="Maroon">D2</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="E15:29_1_4"/>
 ex <font color="Olive">X1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> ex <font color="Olive">Y1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">XV</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">X1</font>,<font color="Olive">Y1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">X1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="borsuk_3.html#E3:29"><i><font color="Green">E31</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:29_1"/><b>then </b>
<font color="Maroon">P'</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<b>hence </b><a NAME="E22:29_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><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="E7:29"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">XV</font></span><a href="borsuk_1.html#K5">:]</a></span> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_3.html#T2" target="_self">Th2</a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>


</div>
