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<b>let </b><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font>;<br/><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="c2">X</font>;<br/>







<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>

<a NAME="E2:69"/><i><font color="Green" title="E55">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</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="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Y</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" target="_self" title="BORSUK_1:def.5">Def5</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">G</font> be    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_1"/>
( <font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">X</font> or <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">X</font>
 ;<br/><div class="add"><a NAME="E2:69_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <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">G</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/><a NAME="E3:69_1_1_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Base-Appr</a> <font color="Maroon" title="c5">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T55" target="_self" title="BORSUK_1:th.55">Th55</a></i>;<br/><a NAME="E4:69_1_1_1"/><b>then </b>
 <a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Base-Appr</a> <font color="Maroon" title="c5">G</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">K</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="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E6:69_1_1_1"/><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c6">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Base-Appr</a> <font color="Maroon" title="c5">G</font>
 <b>and </b><br/><a NAME="E7:69_1_1_1"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c6">K</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>and </b><br/><a NAME="E8:69_1_1_1"/><i><font color="Green" title="E59">A6</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font> holds <br/><font color="Olive" title="b1">c</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T63" target="_self" title="BORSUK_1:th.63">Th63</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">PK</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">Pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">K</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">X</font> ;<br/><a NAME="E10:69_1_1_1"/><i><font color="Green" title="E60">A7</font></i>: 
<font color="Maroon" title="c7">PK</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E7:69_1_1_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="borsuk_1.html#T62" target="_self" title="BORSUK_1:th.62">Th62</a></i>;<br/><a NAME="E11:69_1_1_1"/><i><font color="Green" title="E61">A8</font></i>: 
 <a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Base-Appr</a> <font color="Maroon" title="c5">G</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T51" target="_self" title="BORSUK_1:th.51">Th51</a></i>;<br/><a NAME="E12:69_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">K</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E6:69_1_1_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="tops_2.html#T18" title="TOPS_2:th.18">TOPS_2:18</a></i>;<br/><a NAME="E13:69_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">Pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">K</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Pr2</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">K</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T60" target="_self" title="BORSUK_1:th.60">Th60</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">M</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E15:69_1_1_1"/><i><font color="Green" title="E62">A9</font></i>: 
<font color="Maroon" title="c8">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">Pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">K</font>
 <b>and </b><br/><a NAME="E16:69_1_1_1"/><i><font color="Green" title="E63">A10</font></i>: 
<font color="Maroon" title="c8">M</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c4">A</font>
 <b>and </b><br/><a NAME="E17:69_1_1_1"/><i><font color="Green" title="E64">A11</font></i>: 
<font color="Maroon" title="c8">M</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:69"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="borsuk_1.html#E10:69_1_1_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="compts_1.html#D7" title="COMPTS_1:def.7">COMPTS_1:def 7</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">N</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="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E19:69_1_1_1"/><i><font color="Green" title="E65">A12</font></i>: 
<font color="Maroon" title="c9">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">K</font>
 <b>and </b><br/><a NAME="E20:69_1_1_1"/><i><font color="Green" title="E66">A13</font></i>: 
<font color="Maroon" title="c9">N</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E21:69_1_1_1"/><i><font color="Green" title="E67">A14</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">Pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">M</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E15:69_1_1_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="txt" href="borsuk_1.html#E17:69_1_1_1"><i><font color="Green" title="E64">A11</font></i></a>, <a class="ref" href="borsuk_1.html#T64" target="_self" title="BORSUK_1:th.64">Th64</a></i>;<br/><a NAME="E22:69_1_1_1"/><i><font color="Green" title="E68">A15</font></i>: 
<font color="Maroon" title="c9">N</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">e</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>,<a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a><br/>

<b>assume </b><a NAME="E1:69_1_1_1_1"/><i><font color="Green" title="E69">A16</font></i>: 
<font color="Maroon" title="c10">e</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="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 ;<br/>

<a NAME="E2:69_1_1_1_1"/><b>then </b><i><font color="Green" title="E70">A17</font></i>: 
( <font color="Maroon" title="c10">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c10">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<a NAME="E3:69_1_1_1_1"/><b>then </b><i><font color="Green" title="E71">A18</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:69_1_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c8">M</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E16:69_1_1_1"><i><font color="Green" title="E63">A10</font></i></a>, <a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:69_1_1_1_1"/><i><font color="Green" title="E72">A19</font></i>: 
( <font color="Maroon" title="c10">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">X1</font> &amp; <font color="Maroon" title="c11">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">M</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:69_1_1_1_1"><i><font color="Green" title="E70">A17</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="c12">XY</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="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E8:69_1_1_1_1"/><i><font color="Green" title="E73">A20</font></i>: 
<font color="Maroon" title="c12">XY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N</font>
 <b>and </b><br/><a NAME="E9:69_1_1_1_1"/><i><font color="Green" title="E74">A21</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">Pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">XY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X1</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E21:69_1_1_1"><i><font color="Green" title="E67">A14</font></i></a>, <a class="txt" href="borsuk_1.html#E6:69_1_1_1_1"><i><font color="Green" title="E72">A19</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E10:69_1_1_1_1"/>
<font color="Maroon" title="c12">XY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E19:69_1_1_1"><i><font color="Green" title="E65">A12</font></i></a>, <a class="txt" href="borsuk_1.html#E8:69_1_1_1_1"><i><font color="Green" title="E73">A20</font></i></a></i>;<br/>
<a NAME="E11:69_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">XY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Base-Appr</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E6:69_1_1_1"><i><font color="Green" title="E57">A4</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">X2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c14">Y2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E13:69_1_1_1_1"/><i><font color="Green" title="E75">A22</font></i>: 
<font color="Maroon" title="c12">XY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c13">X2</font>,<font color="Maroon" title="c14">Y2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>and </b><br/><a NAME="E14:69_1_1_1_1"/>
( <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c13">X2</font>,<font color="Maroon" title="c14">Y2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c13">X2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c14">Y2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/>
<a NAME="E15:69_1_1_1_1"/>
<font color="Maroon" title="c12">XY</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E6:69_1_1_1_1"><i><font color="Green" title="E72">A19</font></i></a>, <a class="txt" href="borsuk_1.html#E9:69_1_1_1_1"><i><font color="Green" title="E74">A21</font></i></a>, <a class="ref" href="funct_3.html#T9" title="FUNCT_3:th.9">FUNCT_3:9</a></i>;<br/>
<a NAME="E16:69_1_1_1_1"/><b>then </b><i><font color="Green" title="E76">A23</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X2</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E6:69_1_1_1_1"><i><font color="Green" title="E72">A19</font></i></a>, <a class="txt" href="borsuk_1.html#E9:69_1_1_1_1"><i><font color="Green" title="E74">A21</font></i></a>, <a class="txt" href="borsuk_1.html#E13:69_1_1_1_1"><i><font color="Green" title="E75">A22</font></i></a>, <a class="ref" href="borsuk_1.html#T13" target="_self" title="BORSUK_1:th.13">Th13</a></i>;<br/>
<a NAME="E17:69_1_1_1_1"/>
<font color="Maroon" title="c12">XY</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E8:69_1_1_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="borsuk_1.html#E19:69_1_1_1"><i><font color="Green" title="E65">A12</font></i></a>, <a class="txt" href="borsuk_1.html#E8:69_1_1_1_1"><i><font color="Green" title="E73">A20</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">xy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E19:69_1_1_1_1"/><i><font color="Green" title="E77">A24</font></i>: 
( <font color="Maroon" title="c15">xy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">XY</font> &amp; <font color="Maroon" title="c15">xy</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="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> )
 <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="E20:69_1_1_1_1"/>
<font color="Maroon" title="c15">xy</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</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#E19:69_1_1_1_1"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<a NAME="E21:69_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">xy</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E22:69_1_1_1_1"/><b>then </b><i><font color="Green" title="E78">A25</font></i>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Y2</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E13:69_1_1_1_1"><i><font color="Green" title="E75">A22</font></i></a>, <a class="txt" href="borsuk_1.html#E19:69_1_1_1_1"><i><font color="Green" title="E77">A24</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<a NAME="E23:69_1_1_1_1"/>
<font color="Maroon" title="c10">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c10">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<font color="Maroon" title="c3">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:69_1_1_1_1"><i><font color="Green" title="E69">A16</font></i></a>, <a class="txt" href="borsuk_1.html#E3:69_1_1_1_1"><i><font color="Green" title="E71">A18</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E24:69_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">e</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="c13">X2</font>,<font color="Maroon" title="c14">Y2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E16:69_1_1_1_1"><i><font color="Green" title="E76">A23</font></i></a>, <a class="txt" href="borsuk_1.html#E22:69_1_1_1_1"><i><font color="Green" title="E78">A25</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="E25:69_1_1_1_1"/>
<font color="Maroon" title="c10">e</font> <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="c9">N</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E8:69_1_1_1_1"><i><font color="Green" title="E73">A20</font></i></a>, <a class="txt" href="borsuk_1.html#E13:69_1_1_1_1"><i><font color="Green" title="E75">A22</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E23:69_1_1_1"/><i><font color="Green" title="E69">A26</font></i>: 
<font color="Maroon" title="c9">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Base-Appr</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E6:69_1_1_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="borsuk_1.html#E19:69_1_1_1"><i><font color="Green" title="E65">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green" title="E70">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:69_1_1_1_2"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N</font>
 ;<br/><a NAME="E2:69_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Base-Appr</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E23:69_1_1_1"><i><font color="Green" title="E69">A26</font></i></a></i>;<br/><a NAME="E3:69_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">X1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> ex <font color="Olive" title="b2">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Maroon" title="c10">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="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Olive" title="b1">X1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">Y1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/><b>hence </b><a NAME="E4:69_1_1_1_2"/>
( <font color="Maroon" title="c10">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> &amp;  ex <font color="Olive" title="b1">X1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> ex <font color="Olive" title="b2">Y1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> st <font color="Maroon" title="c10">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="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> )
 ;<br/></div><b>end;</b></div><a NAME="E25:69_1_1_1"/><i><font color="Green" title="E71">A28</font></i>: 
<font color="Maroon" title="c9">N</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E11:69_1_1_1"><i><font color="Green" title="E61">A8</font></i></a>, <a class="txt" href="borsuk_1.html#E23:69_1_1_1"><i><font color="Green" title="E69">A26</font></i></a>, <a class="ref" href="tops_2.html#T18" title="TOPS_2:th.18">TOPS_2:18</a></i>;<br/><a NAME="E26:69_1_1_1"/>
( <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c9">N</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E1:69_1_1_1"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="borsuk_1.html#E22:69_1_1_1"><i><font color="Green" title="E68">A15</font></i></a>, <a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a>, <a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/><a NAME="E27:69_1_1_1"/><b>then </b><i><font color="Green" title="E72">A29</font></i>: 
<font color="Maroon" title="c9">N</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">F</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K11" title="BORSUK_1:func.11">Pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">N</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">X</font> ;<br/><a NAME="E29:69_1_1_1"/>
<font color="Maroon" title="c10">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E25:69_1_1_1"><i><font color="Green" title="E71">A28</font></i></a>, <a class="ref" href="borsuk_1.html#T60" target="_self" title="BORSUK_1:th.60">Th60</a></i>;<br/><a NAME="E30:69_1_1_1"/><b>then </b><i><font color="Green" title="E73">A30</font></i>: 
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">F</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#T26" title="TOPS_2:th.26">TOPS_2:26</a></i>;<br/><a NAME="E31:69_1_1_1"/>
<font color="Maroon" title="c10">F</font> <a href="compts_1.html#R1" title="COMPTS_1:pred.1">is_a_cover_of</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E22:69_1_1_1"><i><font color="Green" title="E68">A15</font></i></a>, <a class="ref" href="borsuk_1.html#T62" target="_self" title="BORSUK_1:th.62">Th62</a></i>;<br/><a NAME="E32:69_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="ref" href="compts_1.html#D1" title="COMPTS_1:def.1">COMPTS_1:def 1</a></i>;<br/><a NAME="E33:69_1_1_1"/><b>then </b>
<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> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E30:69_1_1_1"><i><font color="Green" title="E73">A30</font></i></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">V</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c10">F</font> as    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <font color="Maroon" title="c4">A</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>reconsider </b><font color="Maroon" title="c12">H</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">Pr2</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">N</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">Y</font> ;<br/><a NAME="E36:69_1_1_1"/><i><font color="Green" title="E74">A31</font></i>: 
<font color="Maroon" title="c12">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E20:69_1_1_1"><i><font color="Green" title="E66">A13</font></i></a>, <a class="ref" href="finset_1.html#T17" title="FINSET_1:th.17">FINSET_1:17</a></i>;<br/><a NAME="E37:69_1_1_1"/>
<font color="Maroon" title="c12">H</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E25:69_1_1_1"><i><font color="Green" title="E71">A28</font></i></a>, <a class="ref" href="borsuk_1.html#T60" target="_self" title="BORSUK_1:th.60">Th60</a></i>;<br/><a NAME="E38:69_1_1_1"/><b>then </b><i><font color="Green" title="E75">A32</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c12">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#E36:69_1_1_1"><i><font color="Green" title="E74">A31</font></i></a>, <a class="ref" href="tops_2.html#T27" title="TOPS_2:th.27">TOPS_2:27</a></i>;<br/><a NAME="E39:69_1_1_1"/><i><font color="Green" title="E76">A33</font></i>: 
<font color="Maroon" title="c12">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="borsuk_1.html#E27:69_1_1_1"><i><font color="Green" title="E72">A29</font></i></a>, <a class="ref" href="borsuk_1.html#T61" target="_self" title="BORSUK_1:th.61">Th61</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">C</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:69_1_1_1_3"/>
<font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">H</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">D</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="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E3:69_1_1_1_3"/><i><font color="Green" title="E77">A34</font></i>: 
( <font color="Maroon" title="c14">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N</font> &amp; <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">D</font> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T58" target="_self" title="BORSUK_1:th.58">Th58</a></i>;<br/><a NAME="E4:69_1_1_1_3"/>
<font color="Maroon" title="c14">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E8:69_1_1_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="borsuk_1.html#E19:69_1_1_1"><i><font color="Green" title="E65">A12</font></i></a>, <a class="txt" href="borsuk_1.html#E3:69_1_1_1_3"><i><font color="Green" title="E77">A34</font></i></a></i>;<br/><a NAME="E5:69_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c14">D</font> <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="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">x</font> being   <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="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span><b> such that </b><br/><a NAME="E7:69_1_1_1_3"/><i><font color="Green" title="E78">A35</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font> <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="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E8:69_1_1_1_3"/>
<font color="Maroon" title="c15">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"><font color="Maroon" title="c4">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E7:69_1_1_1_3"><i><font color="Green" title="E78">A35</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="E9:69_1_1_1_3"/><b>then </b><i><font color="Green" title="E79">A36</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c15">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/><a NAME="E10:69_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E11:69_1_1_1_3"/><b>then </b><i><font color="Green" title="E80">A37</font></i>: 
<font color="Maroon" title="c15">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"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<font color="Maroon" title="c3">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:69"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/><a NAME="E12:69_1_1_1_3"/><i><font color="Green" title="E81">A38</font></i>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Y</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E9:69_1_1_1_3"><i><font color="Green" title="E79">A36</font></i></a>, <a class="ref" href="funct_3.html#D6" title="FUNCT_3:def.6">FUNCT_3:def 6</a></i>;<br/><a NAME="E13:69_1_1_1_3"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">D</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E7:69_1_1_1_3"><i><font color="Green" title="E78">A35</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E14:69_1_1_1_3"/>
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">C</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:69"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="borsuk_1.html#E3:69_1_1_1_3"><i><font color="Green" title="E77">A34</font></i></a>, <a class="txt" href="borsuk_1.html#E12:69_1_1_1_3"><i><font color="Green" title="E81">A38</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a>, <a class="txt" href="borsuk_1.html#E11:69_1_1_1_3"><i><font color="Green" title="E80">A37</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E41:69_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c12">H</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E39:69_1_1_1"><i><font color="Green" title="E76">A33</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><a NAME="E42:69_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c12">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E38:69_1_1_1"><i><font color="Green" title="E75">A32</font></i></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">W</font> =  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c12">H</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">t</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/><b>take </b><font color="Maroon" title="c14">V</font> = <font color="Maroon" title="c11">V</font>;<br/><b>take </b><font color="Maroon" title="c15">W</font> = <font color="Maroon" title="c13">W</font>;<br/><b>thus </b><a NAME="E44:69_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">[:</a><span class="default"><font color="Maroon" title="c14">V</font>,<font color="Maroon" title="c15">W</font></span><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E24:69_1_1_1"><i><font color="Green" title="E70">A27</font></i></a>, <a class="ref" href="borsuk_1.html#T56" target="_self" title="BORSUK_1:th.56">Th56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">X</font>
 ;<br/><div class="add"><a NAME="E2:69_1_1_2"/><b>then </b>
<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> <span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">X</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">V</font> =  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c2">X</font> as    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <font color="Maroon" title="c4">A</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>consider </b><font color="Maroon" title="c7">W</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">t</font>;<br/><b>take </b><font color="Maroon" title="c8">V</font> = <font color="Maroon" title="c6">V</font>;<br/><b>take </b><font color="Maroon" title="c9">W</font> = <font color="Maroon" title="c7">W</font>;<br/><a NAME="E5:69_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c9">W</font></span><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/><b>hence </b><a NAME="E6:69_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c9">W</font></span><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:69"/>
 ex <font color="Olive" title="b1">V</font> being   <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <font color="Maroon" title="c4">A</font> ex <font color="Olive" title="b2">W</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">t</font> st <span class="p1"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">[:</a><span class="default"><font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font></span><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 ;<br/>


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