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

<b>let </b><font color="Maroon" title="c5">S</font>, <font color="Maroon" title="c6">T</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c7">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">S</font>;<br/><b>let </b><font color="Maroon" title="c8">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">T</font>;<br/>





<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="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 ;<br/><a NAME="E2:18_1"/>
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="c5">S</font>,<font color="Maroon" title="c6">T</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="c5">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">T</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/><a NAME="E3:18_1"/><b>then </b>
 ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">T</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:18_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">S1</font> = <font color="Maroon" title="c5">S</font>, <font color="Maroon" title="c11">T1</font> = <font color="Maroon" title="c6">T</font> as  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> <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c9">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c10">S1</font>,<font color="Maroon" title="c11">T1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="txt" href="topalg_3.html#E1:18_1"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c13">K</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c12">p</font><b> such that </b><br/><a NAME="E7:18_1"/><i><font color="Green" title="E9">A2</font></i>: 
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c10">S1</font>,<font color="Maroon" title="c11">T1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>  st <font color="Olive" title="b1">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">K</font> holds <br/><span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:18_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="yellow13.html#T17" title="YELLOW13:th.17">YELLOW13:17</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">S1</font>, <font color="Maroon" title="c15">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c11">T1</font><b> such that </b><br/><a NAME="E9:18_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">p2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></i>;<br/><a NAME="E10:18_1"/>
for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">S1</font>  st <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c14">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">S1</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c16">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:18_1_1"/><i><font color="Green" title="E12">A5</font></i>: 
<font color="Maroon" title="c14">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">G</font>
 ;<br/>

<a NAME="E3:18_1_1"/><i><font color="Green" title="E13">A6</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">p2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c16">G</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c11">T1</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="topalg_3.html#E2:18_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:18_1_1"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c16">G</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c11">T1</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E1:18_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c10">S1</font>,<font color="Maroon" title="c11">T1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E6:18_1_1"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c17">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">K</font>
 <b>and </b><br/><a NAME="E7:18_1_1"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c17">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c16">G</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c11">T1</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="topalg_3.html#E9:18_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="topalg_3.html#E3:18_1_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E8:18_1_1"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">V</font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E7:18_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="topalg_3.html#E6:18_1_1"><i><font color="Green" title="E14">A7</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:18_1_1"/><i><font color="Green" title="E16">A9</font></i>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> &amp; <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">V</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E11:18_1_1"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</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="c16">G</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c11">T1</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="topalg_3.html#E7:18_1_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="topalg_3.html#E10:18_1_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:18_1_1"/><b>then </b>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">G</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c11">T1</font></span>)</span></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></i>;<br/>
<a NAME="E13:18_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">G</font> &amp; <font color="Olive" title="b2">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c11">T1</font></span>)</span> &amp; <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E14:18_1_1"/>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c16">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:18_1"/><b>then </b><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c14">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/><a NAME="E12:18_1"/>
for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c11">T1</font>  st <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c15">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c11">T1</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_1_2"/><i><font color="Green" title="E12">A11</font></i>: 
<font color="Maroon" title="c16">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:18_1_2"/><i><font color="Green" title="E13">A12</font></i>: 
<font color="Maroon" title="c15">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">G</font>
 ;<br/>

<a NAME="E3:18_1_2"/><i><font color="Green" title="E14">A13</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">p2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">S1</font></span>)</span>,<font color="Maroon" title="c16">G</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E2:18_1_2"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:18_1_2"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">S1</font></span>)</span>,<font color="Maroon" title="c16">G</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E1:18_1_2"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c10">S1</font>,<font color="Maroon" title="c11">T1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E6:18_1_2"/><i><font color="Green" title="E15">A14</font></i>: 
<font color="Maroon" title="c17">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">K</font>
 <b>and </b><br/><a NAME="E7:18_1_2"/><i><font color="Green" title="E16">A15</font></i>: 
<font color="Maroon" title="c17">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">S1</font></span>)</span>,<font color="Maroon" title="c16">G</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E9:18_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="topalg_3.html#E3:18_1_2"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E8:18_1_2"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">V</font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E7:18_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="topalg_3.html#E6:18_1_2"><i><font color="Green" title="E15">A14</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:18_1_2"/><i><font color="Green" title="E17">A16</font></i>: 
( <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> &amp; <font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">V</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E11:18_1_2"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</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"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">S1</font></span>)</span>,<font color="Maroon" title="c16">G</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E7:18_1_2"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="topalg_3.html#E10:18_1_2"><i><font color="Green" title="E17">A16</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:18_1_2"/><b>then </b>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">S1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">G</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></i>;<br/>
<a NAME="E13:18_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">a2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">S1</font></span>)</span> &amp; <font color="Olive" title="b2">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">G</font> &amp; <font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E14:18_1_2"/>
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c16">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:18_1"/><b>then </b>
<font color="Maroon" title="c15">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">B</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/><b>hence </b><a NAME="E14:18_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">B</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="topalg_3.html#E9:18_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="topalg_3.html#E11:18_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">x</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="E2:18"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">B</font></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x1</font>, <font color="Maroon" title="c11">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:18"/><i><font color="Green" title="E8">A17</font></i>: 
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A</font>
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green" title="E9">A18</font></i>: 
<font color="Maroon" title="c11">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">B</font>
 <b>and </b><br/><a NAME="E6:18"/><i><font color="Green" title="E10">A19</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x1</font>,<font color="Maroon" title="c11">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">S1</font> = <font color="Maroon" title="c5">S</font>, <font color="Maroon" title="c13">T1</font> = <font color="Maroon" title="c6">T</font> as  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> <b>by </b><i><a class="txt" href="topalg_3.html#E4:18"><i><font color="Green" title="E8">A17</font></i></a>, <a class="txt" href="topalg_3.html#E5:18"><i><font color="Green" title="E9">A18</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x1</font> = <font color="Maroon" title="c10">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c12">S1</font> <b>by </b><i><a class="txt" href="topalg_3.html#E4:18"><i><font color="Green" title="E8">A17</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x2</font> = <font color="Maroon" title="c11">x2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c13">T1</font> <b>by </b><i><a class="txt" href="topalg_3.html#E5:18"><i><font color="Green" title="E9">A18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">K1</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c14">x1</font><b> such that </b><br/><a NAME="E11:18"/><i><font color="Green" title="E11">A20</font></i>: 
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">S1</font>  st <font color="Olive" title="b1">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">K1</font> holds <br/><font color="Maroon" title="c7">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E4:18"><i><font color="Green" title="E8">A17</font></i></a>, <a class="ref" href="yellow13.html#T17" title="YELLOW13:th.17">YELLOW13:17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">K2</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c15">x2</font><b> such that </b><br/><a NAME="E13:18"/><i><font color="Green" title="E12">A21</font></i>: 
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c13">T1</font>  st <font color="Olive" title="b1">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">K2</font> holds <br/><font color="Maroon" title="c8">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E5:18"><i><font color="Green" title="E9">A18</font></i></a>, <a class="ref" href="yellow13.html#T17" title="YELLOW13:th.17">YELLOW13:17</a></i>;<br/>
<a NAME="E14:18"/>
for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c12">S1</font>,<font color="Maroon" title="c13">T1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>  st <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/><span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c12">S1</font>,<font color="Maroon" title="c13">T1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_2"/><i><font color="Green" title="E13">A22</font></i>: 
<font color="Maroon" title="c18">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:18_2"/><i><font color="Green" title="E14">A23</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c19">F</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c12">S1</font>,<font color="Maroon" title="c13">T1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E4:18_2"/><i><font color="Green" title="E15">A24</font></i>: 
<font color="Maroon" title="c18">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c19">F</font>
 <b>and </b><br/><a NAME="E5:18_2"/><i><font color="Green" title="E16">A25</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">F</font> holds <br/> ex <font color="Olive" title="b2">X1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">S1</font> ex <font color="Olive" title="b3">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c13">T1</font> st <br/>( <font color="Olive" title="b1">e</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="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> &amp; <font color="Olive" title="b2">X1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">Y1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:18_2"><i><font color="Green" title="E13">A22</font></i></a>, <a class="ref" href="borsuk_1.html#T45" title="BORSUK_1:th.45">BORSUK_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:18_2"/><i><font color="Green" title="E17">A26</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c14">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Z</font>
 <b>and </b><br/><a NAME="E8:18_2"/><i><font color="Green" title="E18">A27</font></i>: 
<font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">F</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E2:18_2"><i><font color="Green" title="E14">A23</font></i></a>, <a class="txt" href="topalg_3.html#E4:18_2"><i><font color="Green" title="E15">A24</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">X1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">S1</font>, <font color="Maroon" title="c22">Y1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c13">T1</font><b> such that </b><br/><a NAME="E10:18_2"/><i><font color="Green" title="E19">A28</font></i>: 
<font color="Maroon" title="c20">Z</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="c21">X1</font>,<font color="Maroon" title="c22">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>and </b><br/><a NAME="E11:18_2"/><i><font color="Green" title="E20">A29</font></i>: 
<font color="Maroon" title="c21">X1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E12:18_2"/><i><font color="Green" title="E21">A30</font></i>: 
<font color="Maroon" title="c22">Y1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E5:18_2"><i><font color="Green" title="E16">A25</font></i></a>, <a class="txt" href="topalg_3.html#E8:18_2"><i><font color="Green" title="E18">A27</font></i></a></i>;<br/>
<a NAME="E13:18_2"/>
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">X1</font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E7:18_2"><i><font color="Green" title="E17">A26</font></i></a>, <a class="txt" href="topalg_3.html#E10:18_2"><i><font color="Green" title="E19">A28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">S1</font><b> such that </b><br/><a NAME="E15:18_2"/><i><font color="Green" title="E22">A31</font></i>: 
<font color="Maroon" title="c23">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">K1</font>
 <b>and </b><br/><a NAME="E16:18_2"/><i><font color="Green" title="E23">A32</font></i>: 
<font color="Maroon" title="c23">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">X1</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E11:18_2"><i><font color="Green" title="E20">A29</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E17:18_2"/>
<font color="Maroon" title="c7">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c23">V1</font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E11:18"><i><font color="Green" title="E11">A20</font></i></a>, <a class="txt" href="topalg_3.html#E15:18_2"><i><font color="Green" title="E22">A31</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E19:18_2"/><i><font color="Green" title="E24">A33</font></i>: 
<font color="Maroon" title="c24">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <b>and </b><br/><a NAME="E20:18_2"/><i><font color="Green" title="E25">A34</font></i>: 
<font color="Maroon" title="c24">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">V1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E21:18_2"/>
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">Y1</font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E7:18_2"><i><font color="Green" title="E17">A26</font></i></a>, <a class="txt" href="topalg_3.html#E10:18_2"><i><font color="Green" title="E19">A28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c25">V2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c13">T1</font><b> such that </b><br/><a NAME="E23:18_2"/><i><font color="Green" title="E26">A35</font></i>: 
<font color="Maroon" title="c25">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">K2</font>
 <b>and </b><br/><a NAME="E24:18_2"/><i><font color="Green" title="E27">A36</font></i>: 
<font color="Maroon" title="c25">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">Y1</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E12:18_2"><i><font color="Green" title="E21">A30</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E25:18_2"/>
<font color="Maroon" title="c8">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c25">V2</font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E13:18"><i><font color="Green" title="E12">A21</font></i></a>, <a class="txt" href="topalg_3.html#E23:18_2"><i><font color="Green" title="E26">A35</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c26">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E27:18_2"/><i><font color="Green" title="E28">A37</font></i>: 
<font color="Maroon" title="c26">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 <b>and </b><br/><a NAME="E28:18_2"/><i><font color="Green" title="E29">A38</font></i>: 
<font color="Maroon" title="c26">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">V2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E29:18_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a1</font>,<font color="Maroon" title="c26">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Z</font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E10:18_2"><i><font color="Green" title="E19">A28</font></i></a>, <a class="txt" href="topalg_3.html#E16:18_2"><i><font color="Green" title="E23">A32</font></i></a>, <a class="txt" href="topalg_3.html#E20:18_2"><i><font color="Green" title="E25">A34</font></i></a>, <a class="txt" href="topalg_3.html#E24:18_2"><i><font color="Green" title="E27">A36</font></i></a>, <a class="txt" href="topalg_3.html#E28:18_2"><i><font color="Green" title="E29">A38</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E30:18_2"/><b>then </b><i><font color="Green" title="E30">A39</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a1</font>,<font color="Maroon" title="c26">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c19">F</font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E8:18_2"><i><font color="Green" title="E18">A27</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E31:18_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">a1</font>,<font color="Maroon" title="c26">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E19:18_2"><i><font color="Green" title="E24">A33</font></i></a>, <a class="txt" href="topalg_3.html#E27:18_2"><i><font color="Green" title="E28">A37</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E32:18_2"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c18">G</font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E4:18_2"><i><font color="Green" title="E15">A24</font></i></a>, <a class="txt" href="topalg_3.html#E30:18_2"><i><font color="Green" title="E30">A39</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:18"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c8">B</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E6:18"><i><font color="Green" title="E10">A19</font></i></a>, <a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/>


</div>
