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

<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c4">T2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c5">R</font> be    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/><b>let </b><font color="Maroon" title="c6">R1</font> be    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>;<br/><b>let </b><font color="Maroon" title="c7">R2</font> be    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green" title="E27">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green" title="E28">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font>
 ;<br/>

<a NAME="E3:61"/><i><font color="Green" title="E29">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">BS</font> =  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S2</font> as    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c6">R1</font> <b>by </b><i><a class="txt" href="yellow12.html#E1:61"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="yellow_9.html#T60" title="YELLOW_9:th.60">YELLOW_9:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">BT</font> =  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">T2</font> as    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c7">R2</font> <b>by </b><i><a class="txt" href="yellow12.html#E2:61"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="yellow_9.html#T60" title="YELLOW_9:th.60">YELLOW_9:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">Bpr</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">R1</font>, <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">R2</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">BS</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">BT</font> ) </span>}</span>  as    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c6">R1</font>,<font color="Maroon" title="c7">R2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="ref" href="yellow_9.html#T40" title="YELLOW_9:th.40">YELLOW_9:40</a></i>;<br/>
<i><font color="Green" title="E30">A4</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">R</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
<b>by </b><i><a class="ref" href="yellow_9.html#D6" title="YELLOW_9:def.6">YELLOW_9:def 6</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="yellow12.html#E3:61"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="yellow12.html#E1:61"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="yellow12.html#E2:61"><i><font color="Green" title="E28">A2</font></i></a></i>
;<br/>
<i><font color="Green" title="E31">A5</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">R1</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>
<b>by </b><i><a class="ref" href="yellow_9.html#D6" title="YELLOW_9:def.6">YELLOW_9:def 6</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>
<b>by </b><i><a class="txt" href="yellow12.html#E1:61"><i><font color="Green" title="E27">A1</font></i></a></i>
;<br/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">R2</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font>
<b>by </b><i><a class="ref" href="yellow_9.html#D6" title="YELLOW_9:def.6">YELLOW_9:def 6</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font>
<b>by </b><i><a class="txt" href="yellow12.html#E2:61"><i><font color="Green" title="E28">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E10:61"/><i><font color="Green" title="E32">A6</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c6">R1</font>,<font color="Maroon" title="c7">R2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="yellow12.html#E7:61"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="yellow12.html#E8:61"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>

<b>set </b><font color="Maroon" title="c11">C</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ;<br/>
<a NAME="E11:61"/><i><font color="Green" title="E33">A7</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span>  is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c5">R</font>
 
<b>by </b><i><a class="txt" href="yellow12.html#E1:61"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="yellow12.html#E2:61"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="yellow12.html#T48" target="_self" title="YELLOW12:th.48">Th48</a></i>;<br/>
<a NAME="E12:61"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Bpr</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_4"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c12">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:61_4_1"/>
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c13">U1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c14">U2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c15">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c16">V2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font><b> such that </b><br/><a NAME="E3:61_4_1"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c13">U1</font>,<font color="Maroon" title="c15">V1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c14">U2</font>,<font color="Maroon" title="c16">V2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>and </b><br/><a NAME="E4:61_4_1"/><i><font color="Green" title="E35">A9</font></i>: 
( <font color="Maroon" title="c13">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c14">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c15">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c16">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/><a NAME="E5:61_4_1"/><i><font color="Green" title="E36">A10</font></i>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">U1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">U2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">V2</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="yellow12.html#E3:61_4_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></i>;<br/><a NAME="E6:61_4_1"/>
( <font color="Maroon" title="c13">U1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S1</font> &amp; <font color="Maroon" title="c14">U2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S2</font> &amp; <font color="Maroon" title="c15">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T1</font> &amp; <font color="Maroon" title="c16">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">T2</font> )
 <b>by </b><i><a class="txt" href="yellow12.html#E4:61_4_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E7:61_4_1"/><b>then </b>
( <font color="Maroon" title="c13">U1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">U2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">BS</font> &amp; <font color="Maroon" title="c15">V1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">BT</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:61_4_1"/>
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Bpr</font>
 <b>by </b><i><a class="txt" href="yellow12.html#E5:61_4_1"><i><font color="Green" title="E36">A10</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c12">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:61_4"/>
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Bpr</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">R1</font>, <font color="Maroon" title="c14">b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">R2</font><b> such that </b><br/><a NAME="E4:61_4"/><i><font color="Green" title="E34">A11</font></i>: 
( <font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> &amp; <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">BS</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">BT</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c15">a1</font>, <font color="Maroon" title="c16">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:61_4"/><i><font color="Green" title="E35">A12</font></i>: 
( <font color="Maroon" title="c15">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">S1</font> &amp; <font color="Maroon" title="c16">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S2</font> &amp; <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">a2</font> )
 <b>by </b><i><a class="txt" href="yellow12.html#E4:61_4"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">b1</font>, <font color="Maroon" title="c18">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:61_4"/><i><font color="Green" title="E36">A13</font></i>: 
( <font color="Maroon" title="c17">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T1</font> &amp; <font color="Maroon" title="c18">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">T2</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c18">b2</font> )
 <b>by </b><i><a class="txt" href="yellow12.html#E4:61_4"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">a1</font> = <font color="Maroon" title="c15">a1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="txt" href="yellow12.html#E6:61_4"><i><font color="Green" title="E35">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">a2</font> = <font color="Maroon" title="c16">a2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font> <b>by </b><i><a class="txt" href="yellow12.html#E6:61_4"><i><font color="Green" title="E35">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">b1</font> = <font color="Maroon" title="c17">b1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font> <b>by </b><i><a class="txt" href="yellow12.html#E8:61_4"><i><font color="Green" title="E36">A13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">b2</font> = <font color="Maroon" title="c18">b2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> <b>by </b><i><a class="txt" href="yellow12.html#E8:61_4"><i><font color="Green" title="E36">A13</font></i></a></i>;<br/>
<a NAME="E13:61_4"/><i><font color="Green" title="E37">A14</font></i>: 
( <font color="Maroon" title="c19">a1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c20">a2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c21">b1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c22">b2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="yellow12.html#E6:61_4"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="yellow12.html#E8:61_4"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E14:61_4"/>
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c19">a1</font>,<font color="Maroon" title="c21">b1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c20">a2</font>,<font color="Maroon" title="c22">b2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="yellow12.html#E4:61_4"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="yellow12.html#E6:61_4"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="yellow12.html#E8:61_4"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></i>;<br/>
<b>hence </b><a NAME="E15:61_4"/>
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow12.html#E13:61_4"><i><font color="Green" title="E37">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:61"/>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c6">R1</font>,<font color="Maroon" title="c7">R2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="yellow12.html#E10:61"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="yellow12.html#E11:61"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="yellow_9.html#T25" title="YELLOW_9:th.25">YELLOW_9:25</a></i>;<br/>


</div>
