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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V1">lower-bounded</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V1">lower-bounded</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">S'</font> be  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V1" target="_self">lower</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">T'</font> be  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V1" target="_self">lower</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>;<br/>





<b>set </b><font color="Maroon">SxT</font> = <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">BS</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S'</font> : verum </span>}</span>  as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">S'</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">BT</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T'</font> : verum </span>}</span>  as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T'</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">B1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T'</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BT</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T'</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S'</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BS</font> </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">BB</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T'</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BT</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T'</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S'</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BS</font> </span>}</span>  as    <a href="cantor_1.html#M2">prebasis</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="yellow_9.html#T41">YELLOW_9:41</a></i>;<br/>
<b>consider </b><font color="Maroon">ST</font> being  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V1" target="_self">lower</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">BX</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ST</font> : verum </span>}</span>  as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">ST</font> <b>by </b><i/>;<br/>
<a NAME="E6:24"/><i><font color="Green">E30</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S'</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T'</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T'</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ST</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">ST</font> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span> )
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E7:24"/><b>then </b><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ST</font> <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">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E8:24"/><i><font color="Green">E36</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">S'</font>,<font color="Maroon">T'</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">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</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="E9:24"/>
<font color="Maroon">BB</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">BX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:24_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T'</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BT</font> </span>}</span>  or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T'</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S'</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BS</font> </span>}</span>  )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font>,<font color="Olive">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T'</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BT</font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T'</font><b> such that </b><br/><a NAME="E3:24_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">z</font> <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">S'</font>,<font color="Maroon">b</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BT</font> )
 ;<br/><b>consider </b><font color="Maroon">t'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T'</font><b> such that </b><br/><a NAME="E5:24_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">t'</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/><a NAME="E7:24_1_1_1"/><i><font color="Green">E63</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">t'</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ST</font> <b>by </b><i/>;<br/><a NAME="E9:24_1_1_1"/><i><font color="Green">E65</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E10:24_1_1_1"/>
 <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="waybel14.html#T10">WAYBEL14:10</a></i>;<br/><a NAME="E11:24_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span></span>)</span> <a href="subset_1.html#K3">`</a>  = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</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">S</font>,<font color="Maroon">b</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, , , </i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</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">S</font>,<font color="Maroon">b</font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E13:24_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BX</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Olive">a</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T'</font></span><a href="zfmisc_1.html#K2">:]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S'</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BS</font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S'</font><b> such that </b><br/><a NAME="E3:24_1_1_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T'</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BS</font> )
 ;<br/><b>consider </b><font color="Maroon">s'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S'</font><b> such that </b><br/><a NAME="E5:24_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">s'</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E7:24_1_1_2"/><i><font color="Green">E74</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">s'</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ST</font> <b>by </b><i/>;<br/><a NAME="E9:24_1_1_2"/><i><font color="Green">E76</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E10:24_1_1_2"/>
 <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="waybel14.html#T10">WAYBEL14:10</a></i>;<br/><a NAME="E11:24_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<span class="p3">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">T</font></span>)</span></span><a href="yellow_3.html#K7">]</a></span></span>)</span> <a href="subset_1.html#K3">`</a>  = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</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">S</font>,<a href="xboole_0.html#K1">{}</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">a</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E13:24_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BX</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:24"/><b>then </b>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BX</font>
 
<b>by </b><i>, , <a class="ref" href="cantor_1.html#T16">CANTOR_1:16</a></i>;<br/>
<a NAME="E11:24"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BX</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="cantor_1.html#T9">CANTOR_1:9</a></i>;<br/>
<a NAME="E12:24"/><i><font color="Green">E78</font></i>: 
(  <a href="cantor_1.html#K2">FinMeetCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">ST</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">ST</font> &amp;  <a href="cantor_1.html#K2">FinMeetCl</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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#T5">CANTOR_1:5</a></i>;<br/>
<a NAME="E13:24"/><i><font color="Green">E79</font></i>: 
(  <a href="cantor_1.html#K1">UniCl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">ST</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">ST</font> &amp;  <a href="cantor_1.html#K1">UniCl</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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#T6">CANTOR_1:6</a></i>;<br/>
<a NAME="E14:24"/>
<font color="Maroon">BX</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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:24_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BX</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ST</font><b> such that </b><br/><a NAME="E3:24_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:24_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</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/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">s</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S'</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">t'</font> = <font color="Maroon">t</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T'</font> <b>by </b><i/>;<br/>
<a NAME="E10:24_2"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:24_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K13">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">s</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span><a href="yellow_3.html#K13">:]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="yellow_3.html#K13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">t</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="yellow_3.html#K13">:]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:24_2"/><i><font color="Green">E83</font></i>: 
(  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">s'</font> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">t'</font> )
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> = <span class="p1"><a href="yellow_3.html#K13">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">s</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span><a href="yellow_3.html#K13">:]</a></span>, <font color="Maroon">A2</font> = <span class="p1"><a href="yellow_3.html#K13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">t</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span><a href="yellow_3.html#K13">:]</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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i>, , </i>;<br/>
<a NAME="E14:24_2"/>
( <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">s'</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">BS</font> &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">t'</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">BT</font> &amp; <font color="Maroon">BS</font> is <a href="tops_2.html#V1">open</a> &amp; <font color="Maroon">BT</font> is <a href="tops_2.html#V1">open</a> )
 
<b>by </b><i><a class="ref" href="yellow_9.html#T28">YELLOW_9:28</a></i>;<br/>
<a NAME="E15:24_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">s'</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">t'</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S'</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T'</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T53">TOPS_1:53</a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E16:24_2"/><b>then </b>
( <font color="Maroon">A1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A2</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, , <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>
<a NAME="E17:24_2"/><b>then </b>
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T37">TOPS_1:37</a></i>;<br/>
<b>hence </b><a NAME="E18:24_2"/>
<font color="Maroon">z</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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="waybel19.html#D1" target="_self">Def1</a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:24"/><b>then </b>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BX</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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="cantor_1.html#T16">CANTOR_1:16</a></i>;<br/>
<a NAME="E16:24"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BX</font></span>)</span> <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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="cantor_1.html#T9">CANTOR_1:9</a></i>;<br/>
<a NAME="E17:24"/>
(  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BX</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">ST</font> &amp;  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font> is    <a href="cantor_1.html#M1">Basis</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span> )
 
<b>by </b><i><a class="ref" href="yellow_9.html#T23">YELLOW_9:23</a></i>;<br/>
<a NAME="E18:24"/><b>then </b>
( the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">ST</font> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BX</font></span>)</span> &amp; 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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow_9.html#T22">YELLOW_9:22</a></i>;<br/>
<a NAME="E19:24"/><b>then </b>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">ST</font> <a href="hidden.html#R1">=</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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E20:24"/>
 <a href="waybel19.html#K1" target="_self">omega</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">T</font></span><a href="yellow_3.html#K3">:]</a></span> <a href="hidden.html#R1">=</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">S'</font>,<font color="Maroon">T'</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i/>;<br/>


</div>
