<?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">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V2">compact</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">G</font> be  <a href="pre_topc.html#V3">open</a> <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">X</font>,<font color="Maroon">Y</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:23"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x'</font> where <font color="Olive">x'</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x'</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:23"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x'</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 ;<br/>
<a NAME="E4:23"/><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x'</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T54">BORSUK_1:54</a></i>;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:23_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x'</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:23_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x'</font></span><a href="domain_1.html#K6">}</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:23_1"/><i><font color="Green">E43</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">G</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:23_1"/>
 <a href="borsuk_1.html#K10">Base-Appr</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</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">X</font>, <font color="Olive">Y1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <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="tarski.html#R1">c=</a> <font color="Maroon">G</font> &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> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#D6">BORSUK_1:def 6</a></i>;<br/><b>then consider </b><font color="Maroon">X1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">Y1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E8:23_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">Z</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; <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> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &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="ref" href="borsuk_3.html#T1" target="_self">Th1</a></i>;<br/><b>thus </b><a NAME="E9:23_1"/>
 ex <font color="Olive">G1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ex <font color="Olive">H1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">G1</font>,<font color="Olive">H1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">G1</font>,<font color="Olive">H1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Olive">G1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">H1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="borsuk_3.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">G1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ex <font color="Olive">H1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">G1</font>,<font color="Olive">H1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">G1</font>,<font color="Olive">H1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">G1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">H1</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">G1</font>,<font color="Olive">H1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> );<br/>
<a NAME="E6:23"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> holds <br/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:23_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">G1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">H1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:23_2"/><i><font color="Green">E53</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">H1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">H1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">H1</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i/>;<br/>
<a NAME="E4:23_2"/>
 ex <font color="Olive">G2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ex <font color="Olive">H2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">H1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">G2</font>,<font color="Olive">H2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">G2</font>,<font color="Olive">H2</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">G2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">H2</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">G2</font>,<font color="Olive">H2</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E5:23_2"/>
 ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Olive">j</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:23"/>
 ex <font color="Olive">f</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> st <br/>for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<b>hence </b><a NAME="E8:23"/>
 ex <font color="Olive">f</font> being   <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> st <br/>for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> holds <br/> ex <font color="Olive">G1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ex <font color="Olive">H1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">G1</font>,<font color="Olive">H1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">G1</font>,<font color="Olive">H1</font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">G1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">H1</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">G1</font>,<font color="Olive">H1</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 ;<br/>


</div>
