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<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">XX</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D13" target="_self" title="BORSUK_1:def.13">BORSUK_1:def 13</a><br/>

<b>assume </b><a NAME="E1:93_1_1"/><i><font color="Green" title="E71">A1</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">V</font> be    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <font color="Maroon" title="c4">A</font>;<br/>

<a NAME="E2:93_1_1"/><i><font color="Green" title="E72">A2</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<a NAME="E3:93_1_1"/><i><font color="Green" title="E73">A3</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D2" title="CONNSP_2:def.2">CONNSP_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:93_1_1_1_1"/>
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> or  ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font> st <br/>( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="borsuk_1.html#E1:93_1_1"><i><font color="Green" title="E71">A1</font></i></a>, <a class="ref" href="borsuk_1.html#T74" target="_self" title="BORSUK_1:th.74">Th74</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:93_1_1_1_1_1"/><i><font color="Green" title="E74">A4</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">C</font> = <font color="Maroon" title="c4">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> ;<br/><b>reconsider </b><font color="Maroon" title="c7">E</font> = <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font></span>)</span> <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="c2">X</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> ;<br/><a NAME="E4:93_1_1_1_1_1"/><i><font color="Green" title="E75">A5</font></i>: 
<font color="Maroon" title="c7">E</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/><a NAME="E5:93_1_1_1_1_1"/>
<font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:93_1_1"><i><font color="Green" title="E73">A3</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/><a NAME="E6:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">E</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:93_1_1_1_1_1"><i><font color="Green" title="E75">A5</font></i></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/><a NAME="E7:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">E</font> is    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <font color="Maroon" title="c6">C</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D2" title="CONNSP_2:def.2">CONNSP_2:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E9:93_1_1_1_1_1"/><i><font color="Green" title="E76">A6</font></i>: 
<font color="Maroon" title="c8">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E10:93_1_1_1_1_1"/><i><font color="Green" title="E77">A7</font></i>: 
( <font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W</font> &amp; <font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">E</font> )
 <b>and </b><br/><a NAME="E11:93_1_1_1_1_1"/><i><font color="Green" title="E78">A8</font></i>: 
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="c2">X</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b1">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">W</font> holds <br/><font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:93_1_1_1_1_1"><i><font color="Green" title="E74">A4</font></i></a>, <a class="ref" href="borsuk_1.html#D13" target="_self" title="BORSUK_1:def.13">Def13</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">XX</font><b> such that </b><br/><a NAME="E13:93_1_1_1_1_1"/><i><font color="Green" title="E79">A9</font></i>: 
<font color="Maroon" title="c9">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E14:93_1_1_1_1_1"/><i><font color="Green" title="E80">A10</font></i>: 
<font color="Maroon" title="c8">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G</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="c2">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E9:93_1_1_1_1_1"><i><font color="Green" title="E76">A6</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/><b>take </b><font color="Maroon" title="c10">H</font> = <font color="Maroon" title="c9">G</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font></span>)</span>;<br/><b>thus </b><a NAME="E15:93_1_1_1_1_1"/>
<font color="Maroon" title="c10">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E13:93_1_1_1_1_1"><i><font color="Green" title="E79">A9</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/><a NAME="E16:93_1_1_1_1_1"/><i><font color="Green" title="E81">A11</font></i>: 
<font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">G</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E14:93_1_1_1_1_1"><i><font color="Green" title="E80">A10</font></i></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E17:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">G</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E10:93_1_1_1_1_1"><i><font color="Green" title="E77">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E18:93_1_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">H</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:93_1_1"><i><font color="Green" title="E73">A3</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/><a NAME="E19:93_1_1_1_1_1"/>
<font color="Maroon" title="c10">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><b>hence </b><a NAME="E20:93_1_1_1_1_1"/>
<font color="Maroon" title="c10">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:93_1_1"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E21:93_1_1_1_1_1"/>
<font color="Maroon" title="c7">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E22:93_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E10:93_1_1_1_1_1"><i><font color="Green" title="E77">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E23:93_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E82">A12</font></i>: 
<font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">H</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E16:93_1_1_1_1_1"><i><font color="Green" title="E81">A11</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/><b>let </b><font color="Maroon" title="c11">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">XX</font>;<br/><b>assume </b><a NAME="E24:93_1_1_1_1_1"/><i><font color="Green" title="E83">A13</font></i>: 
( <font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c11">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">H</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:93_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> or  ex <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font> st <br/>( not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )
 </span><b>by </b><i><a class="txt" href="borsuk_1.html#E24:93_1_1_1_1_1"><i><font color="Green" title="E83">A13</font></i></a>, <a class="ref" href="borsuk_1.html#T74" target="_self" title="BORSUK_1:th.74">Th74</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:93_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E84">A14</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 ;<br/><div class="add"><a NAME="E2:93_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c9">G</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E24:93_1_1_1_1_1"><i><font color="Green" title="E83">A13</font></i></a>, <a class="ref" href="xboole_1.html#T74" title="XBOOLE_1:th.74">XBOOLE_1:74</a></i>;<br/><a NAME="E3:93_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">W</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E14:93_1_1_1_1_1"><i><font color="Green" title="E80">A10</font></i></a>, <a class="txt" href="borsuk_1.html#E1:93_1_1_1_1_1_1_1_1"><i><font color="Green" title="E84">A14</font></i></a>, <a class="ref" href="xboole_1.html#T77" title="XBOOLE_1:th.77">XBOOLE_1:77</a></i>;<br/><a NAME="E4:93_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E11:93_1_1_1_1_1"><i><font color="Green" title="E78">A8</font></i></a>, <a class="txt" href="borsuk_1.html#E1:93_1_1_1_1_1_1_1_1"><i><font color="Green" title="E84">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E5:93_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">H</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E23:93_1_1_1_1_1"><i><font color="Green" title="E82">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:93_1_1_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font> st <br/>( not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font><b> such that </b><br/><a NAME="E3:93_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E84">A15</font></i>: 
( not <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 ;<br/><a NAME="E4:93_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">H</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E24:93_1_1_1_1_1"><i><font color="Green" title="E83">A13</font></i></a>, <a class="txt" href="borsuk_1.html#E3:93_1_1_1_1_1_1_1_2"><i><font color="Green" title="E84">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></i>;<br/><b>hence </b><a NAME="E5:93_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">H</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:93_1_1_1_1_1_1_1_2"><i><font color="Green" title="E84">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E26:93_1_1_1_1_1"/>
<font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">H</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:93_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font> st <br/>( not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font><b> such that </b><br/><a NAME="E3:93_1_1_1_1_2"/><i><font color="Green" title="E74">A16</font></i>: 
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E4:93_1_1_1_1_2"/><i><font color="Green" title="E75">A17</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><a NAME="E5:93_1_1_1_1_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">XX</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">C</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">XX</font> ;<br/><b>take </b><font color="Maroon" title="c8">W</font> = <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">C</font>;<br/><a NAME="E7:93_1_1_1_1_2"/>
<font color="Maroon" title="c7">C</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D14" target="_self" title="BORSUK_1:def.14">Def14</a></i>;<br/><a NAME="E8:93_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/><a NAME="E9:93_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/><b>hence </b><a NAME="E10:93_1_1_1_1_2"/>
<font color="Maroon" title="c8">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="subset_1.html#T32" title="SUBSET_1:th.32">SUBSET_1:32</a></i>;<br/><a NAME="E11:93_1_1_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:93_1_1_1_1_2"><i><font color="Green" title="E75">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E12:93_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:93_1_1"><i><font color="Green" title="E73">A3</font></i></a>, <a class="txt" href="borsuk_1.html#E3:93_1_1_1_1_2"><i><font color="Green" title="E74">A16</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E13:93_1_1_1_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E4:93_1_1_1_1_2"><i><font color="Green" title="E75">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E14:93_1_1_1_1_2"/>
<font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><b>hence </b><a NAME="E15:93_1_1_1_1_2"/>
<font color="Maroon" title="c8">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:93_1_1"><i><font color="Green" title="E72">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>let </b><font color="Maroon" title="c9">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">XX</font>;<br/><b>assume </b><a NAME="E16:93_1_1_1_1_2"/><i><font color="Green" title="E76">A18</font></i>: 
( <font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c9">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">W</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:93_1_1_1_1_2_1"/><i><font color="Green" title="E77">A19</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 ;<br/><a NAME="E2:93_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/><b>hence </b><a NAME="E3:93_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_1.html#E16:93_1_1_1_1_2"><i><font color="Green" title="E76">A18</font></i></a>, <a class="txt" href="borsuk_1.html#E1:93_1_1_1_1_2_1"><i><font color="Green" title="E77">A19</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c10">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font><b> such that </b><br/><a NAME="E19:93_1_1_1_1_2"/>
not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E20:93_1_1_1_1_2"/><i><font color="Green" title="E77">A20</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E16:93_1_1_1_1_2"><i><font color="Green" title="E76">A18</font></i></a>, <a class="ref" href="borsuk_1.html#T74" target="_self" title="BORSUK_1:th.74">Th74</a></i>;<br/><a NAME="E21:93_1_1_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E16:93_1_1_1_1_2"><i><font color="Green" title="E76">A18</font></i></a>, <a class="txt" href="borsuk_1.html#E20:93_1_1_1_1_2"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#T56" title="ZFMISC_1:th.56">ZFMISC_1:56</a></i>;<br/><b>hence </b><a NAME="E22:93_1_1_1_1_2"/>
<font color="Maroon" title="c9">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E20:93_1_1_1_1_2"><i><font color="Green" title="E77">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:93_1_1"/>
 ex <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">XX</font> st <br/>( <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font> &amp; ( 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="c1">XX</font>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K15" title="BORSUK_1:func.15">TrivExt</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">W</font> holds <br/><font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">W</font> ) )
 ;<br/>


</div>
