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

<span class="kw">let </span><font color="Maroon" title="c4">S</font>, <font color="Maroon" title="c5">T</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">S</font><br/> for <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T</font> holds   <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="Olive" title="b1">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="Olive" title="b2">B</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T</font> holds   <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Olive" title="b1">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c6">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="Olive" title="b1">B</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c7">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c6">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="c7">B</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/>





<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><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> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c6">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="c7">B</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c6">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="c7">B</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:18_1"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c8">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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c6">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="c7">B</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">S1</font> = <font color="Maroon" title="c4">S</font>, <font color="Maroon" title="c10">T1</font> = <font color="Maroon" title="c5">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> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">p</font> = <font color="Maroon" title="c8">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c9">S1</font>,<font color="Maroon" title="c10">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">K</font> being   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c11">p</font><span class="kw"> such that </span><br/><a NAME="E5:18_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c9">S1</font>,<font color="Maroon" title="c10">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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="c12">K</font> holds <br/><span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="yellow13.html#T17" title="YELLOW13:th.17">YELLOW13:17</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c9">S1</font>, <font color="Maroon" title="c14">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c10">T1</font><span class="kw"> such that </span><br/><a NAME="E7:18_1"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c13">p1</font>,<font color="Maroon" title="c14">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T50" title="BORSUK_1:th.50">BORSUK_1:50</a></span>;<br/><a NAME="E8:18_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">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="c14">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="c7">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_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c10">T1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c14">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">G</font> implies <font color="Maroon" title="c7">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_1_1"/>
( <font color="Maroon" title="c15">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c14">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">G</font></span><br/>

<a NAME="E2:18_1_1"/><span class="kw">then </span>
( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c13">p1</font>,<font color="Maroon" title="c14">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c9">S1</font></span>)</span>,<font color="Maroon" title="c15">G</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c9">S1</font></span>)</span>,<font color="Maroon" title="c15">G</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c9">S1</font>,<font color="Maroon" title="c10">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E4:18_1_1"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c16">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">K</font>
 <span class="kw">and </span><br/><a NAME="E5:18_1_1"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c16">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c9">S1</font></span>)</span>,<font color="Maroon" title="c15">G</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></span>;<br/>
<a NAME="E6:18_1_1"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c16">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E4:18_1_1"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:18_1_1"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E9:18_1_1"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c9">S1</font></span>)</span>,<font color="Maroon" title="c15">G</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_1_1"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E8:18_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E10:18_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">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="c9">S1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">G</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></span>;<br/>
<a NAME="E11:18_1_1"/><span class="kw">then </span>
 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="c6">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="c9">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="c7">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">G</font> &amp; <font color="Maroon" title="c17">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:18_1_1"/>
<font color="Maroon" title="c7">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<font color="Maroon" title="c14">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="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></span>;<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="c9">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="c13">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="c6">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_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c9">S1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c13">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">G</font> implies <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_1_2"/>
( <font color="Maroon" title="c15">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c13">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">G</font></span><br/>

<a NAME="E2:18_1_2"/><span class="kw">then </span>
( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c13">p1</font>,<font color="Maroon" title="c14">p2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c15">G</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">T1</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c15">G</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">T1</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c9">S1</font>,<font color="Maroon" title="c10">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E4:18_1_2"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon" title="c16">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">K</font>
 <span class="kw">and </span><br/><a NAME="E5:18_1_2"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<font color="Maroon" title="c16">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c15">G</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">T1</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></span>;<br/>
<a NAME="E6:18_1_2"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c16">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E4:18_1_2"><span class="lab"><font color="Green" title="E11">A8</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:18_1_2"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">V</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E9:18_1_2"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c15">G</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c10">T1</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_1_2"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E8:18_1_2"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E10:18_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">G</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">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="c10">T1</font></span>)</span></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T123" title="ZFMISC_1:th.123">ZFMISC_1:123</a></span>;<br/>
<a NAME="E11:18_1_2"/><span class="kw">then </span>
 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="c6">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">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="c7">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="c10">T1</font></span>)</span> &amp; <font color="Maroon" title="c17">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:18_1_2"/>
<font color="Maroon" title="c6">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E11:18_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="c6">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E12:18_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c6">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="c7">B</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E9:18_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c6">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="c7">B</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> or <font color="Maroon" title="c8">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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:18"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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="c6">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="c7">B</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c10">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:18"/><span class="lab"><font color="Green" title="E7">A11</font></span>: 
<font color="Maroon" title="c9">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="c6">A</font>
 <span class="kw">and </span><br/><a NAME="E5:18"/><span class="lab"><font color="Green" title="E8">A12</font></span>: 
<font color="Maroon" title="c10">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="c7">B</font>
 <span class="kw">and </span><br/><a NAME="E6:18"/><span class="lab"><font color="Green" title="E9">A13</font></span>: 
<font color="Maroon" title="c8">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="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">S1</font> = <font color="Maroon" title="c4">S</font>, <font color="Maroon" title="c12">T1</font> = <font color="Maroon" title="c5">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E7">A11</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E8">A12</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c9">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c11">S1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E7">A11</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">K1</font> being   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c13">x1</font><span class="kw"> such that </span><br/><a NAME="E10:18"/><span class="lab"><font color="Green" title="E10">A14</font></span>: 
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="c11">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="c14">K1</font> holds <br/><font color="Maroon" title="c6">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E7">A11</font></span></a>, <a class="ref" href="yellow13.html#T17" title="YELLOW13:th.17">YELLOW13:17</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">x2</font> = <font color="Maroon" title="c10">x2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c12">T1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E8">A12</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">K2</font> being   <a href="yellow_8.html#NM1" title="YELLOW_8:NM.1">Basis</a> of <font color="Maroon" title="c15">x2</font><span class="kw"> such that </span><br/><a NAME="E13:18"/><span class="lab"><font color="Green" title="E11">A15</font></span>: 
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">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="c16">K2</font> holds <br/><font color="Maroon" title="c7">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E8">A12</font></span></a>, <a class="ref" href="yellow13.html#T17" title="YELLOW13:th.17">YELLOW13:17</a></span>;<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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c11">S1</font>,<font color="Maroon" title="c12">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c11">S1</font>,<font color="Maroon" title="c12">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">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#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G</font> implies <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:18_2"/><span class="lab"><font color="Green" title="E12">A16</font></span>: 
<font color="Maroon" title="c17">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E2:18_2"/><span class="lab"><font color="Green" title="E13">A17</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c18">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c11">S1</font>,<font color="Maroon" title="c12">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E4:18_2"/><span class="lab"><font color="Green" title="E14">A18</font></span>: 
<font color="Maroon" title="c17">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="c18">F</font>
 <span class="kw">and </span><br/><a NAME="E5:18_2"/><span class="lab"><font color="Green" title="E15">A19</font></span>: 
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="c18">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="c11">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="c12">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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="ref" href="borsuk_1.html#T45" title="BORSUK_1:th.45">BORSUK_1:45</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:18_2"/><span class="lab"><font color="Green" title="E16">A20</font></span>: 
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c15">x2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">Z</font>
 <span class="kw">and </span><br/><a NAME="E8:18_2"/><span class="lab"><font color="Green" title="E17">A21</font></span>: 
<font color="Maroon" title="c19">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_2"><span class="lab"><font color="Green" title="E13">A17</font></span></a>, <a class="txt" href="#E4:18_2"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">X1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c11">S1</font>, <font color="Maroon" title="c21">Y1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">T1</font><span class="kw"> such that </span><br/><a NAME="E10:18_2"/><span class="lab"><font color="Green" title="E18">A22</font></span>: 
<font color="Maroon" title="c19">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c20">X1</font>,<font color="Maroon" title="c21">Y1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E11:18_2"/><span class="lab"><font color="Green" title="E19">A23</font></span>: 
<font color="Maroon" title="c20">X1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E12:18_2"/><span class="lab"><font color="Green" title="E20">A24</font></span>: 
<font color="Maroon" title="c21">Y1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_2"><span class="lab"><font color="Green" title="E15">A19</font></span></a>, <a class="txt" href="#E8:18_2"><span class="lab"><font color="Green" title="E17">A21</font></span></a></span>;<br/>
<a NAME="E13:18_2"/>
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">Y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_2"><span class="lab"><font color="Green" title="E16">A20</font></span></a>, <a class="txt" href="#E10:18_2"><span class="lab"><font color="Green" title="E18">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">V2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">T1</font><span class="kw"> such that </span><br/><a NAME="E15:18_2"/><span class="lab"><font color="Green" title="E21">A25</font></span>: 
<font color="Maroon" title="c22">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">K2</font>
 <span class="kw">and </span><br/><a NAME="E16:18_2"/><span class="lab"><font color="Green" title="E22">A26</font></span>: 
<font color="Maroon" title="c22">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">Y1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:18_2"><span class="lab"><font color="Green" title="E20">A24</font></span></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></span>;<br/>
<a NAME="E17:18_2"/>
<font color="Maroon" title="c7">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c22">V2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:18"><span class="lab"><font color="Green" title="E11">A15</font></span></a>, <a class="txt" href="#E15:18_2"><span class="lab"><font color="Green" title="E21">A25</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E19:18_2"/><span class="lab"><font color="Green" title="E23">A27</font></span>: 
<font color="Maroon" title="c23">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">and </span><br/><a NAME="E20:18_2"/><span class="lab"><font color="Green" title="E24">A28</font></span>: 
<font color="Maroon" title="c23">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">V2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E21:18_2"/>
<font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18_2"><span class="lab"><font color="Green" title="E16">A20</font></span></a>, <a class="txt" href="#E10:18_2"><span class="lab"><font color="Green" title="E18">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c11">S1</font><span class="kw"> such that </span><br/><a NAME="E23:18_2"/><span class="lab"><font color="Green" title="E25">A29</font></span>: 
<font color="Maroon" title="c24">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">K1</font>
 <span class="kw">and </span><br/><a NAME="E24:18_2"/><span class="lab"><font color="Green" title="E26">A30</font></span>: 
<font color="Maroon" title="c24">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:18_2"><span class="lab"><font color="Green" title="E19">A23</font></span></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></span>;<br/>
<a NAME="E25:18_2"/>
<font color="Maroon" title="c6">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c24">V1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:18"><span class="lab"><font color="Green" title="E10">A14</font></span></a>, <a class="txt" href="#E23:18_2"><span class="lab"><font color="Green" title="E25">A29</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E27:18_2"/><span class="lab"><font color="Green" title="E27">A31</font></span>: 
<font color="Maroon" title="c25">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <span class="kw">and </span><br/><a NAME="E28:18_2"/><span class="lab"><font color="Green" title="E28">A32</font></span>: 
<font color="Maroon" title="c25">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<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="c25">a1</font>,<font color="Maroon" title="c23">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="c19">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:18_2"><span class="lab"><font color="Green" title="E18">A22</font></span></a>, <a class="txt" href="#E24:18_2"><span class="lab"><font color="Green" title="E26">A30</font></span></a>, <a class="txt" href="#E28:18_2"><span class="lab"><font color="Green" title="E28">A32</font></span></a>, <a class="txt" href="#E16:18_2"><span class="lab"><font color="Green" title="E22">A26</font></span></a>, <a class="txt" href="#E20:18_2"><span class="lab"><font color="Green" title="E24">A28</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E30:18_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A33</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">a1</font>,<font color="Maroon" title="c23">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="c18">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18_2"><span class="lab"><font color="Green" title="E17">A21</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<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="c25">a1</font>,<font color="Maroon" title="c23">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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:18_2"><span class="lab"><font color="Green" title="E27">A31</font></span></a>, <a class="txt" href="#E19:18_2"><span class="lab"><font color="Green" title="E23">A27</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:18_2"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="txt" href="#E30:18_2"><span class="lab"><font color="Green" title="E29">A33</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:18"/>
<font color="Maroon" title="c8">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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c7">B</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E9">A13</font></span></a>, <a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
