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<span class="kw">let </span><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">X</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <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">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font><br/> for <font color="Olive" title="b2">A</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="b2">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  holds <br/>for <font color="Olive" title="b3">G</font> being   <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">A</font>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>  ex <font color="Olive" title="b4">V</font> being   <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <font color="Olive" title="b2">A</font> ex <font color="Olive" title="b5">W</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">t</font> st <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Olive" title="b4">V</font>,<font color="Olive" title="b5">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">G</font></span><br/><span class="kw">let </span><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>; <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="c2">X</font>  st <font color="Olive" title="b1">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  holds <br/>for <font color="Olive" title="b2">G</font> being   <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <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>,<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#K4" title="BORSUK_1:func.4">:]</a></span>  ex <font color="Olive" title="b3">V</font> being   <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <font color="Olive" title="b1">A</font> ex <font color="Olive" title="b4">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#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Olive" title="b3">V</font>,<font color="Olive" title="b4">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">G</font></span><br/><span class="kw">let </span><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>; <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="c4">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  implies for <font color="Olive" title="b1">G</font> being   <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>  ex <font color="Olive" title="b2">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="b3">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#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Olive" title="b2">V</font>,<font color="Olive" title="b3">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:78"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
<font color="Maroon" title="c4">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</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">G</font> being   <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>  ex <font color="Olive" title="b2">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="b3">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#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Olive" title="b2">V</font>,<font color="Olive" title="b3">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font></span><br/>

<span class="kw">let </span><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#K4" title="BORSUK_1:func.4">[:</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#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">  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#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font></span><br/>

<a NAME="E2:78"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" target="_self" title="BORSUK_1:def.5">Def5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:78_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="78_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:78_1_1_1"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font></span><br/><div class="add"><a NAME="E2:78_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D2" title="CONNSP_2:def.2">CONNSP_2:def 2</a></span>;<br/><a NAME="E3:78_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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#K8" title="BORSUK_1:func.8">Base-Appr</a> <font color="Maroon" title="c5">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T55" target="_self" title="BORSUK_1:th.55">Th55</a></span>;<br/><a NAME="E4:78_1_1_1"/><span class="kw">then </span>
 <a href="borsuk_1.html#K8" title="BORSUK_1:func.8">Base-Appr</a> <font color="Maroon" title="c5">G</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E6:78_1_1_1"/><span class="lab"><font color="Green" title="E53">A4</font></span>: 
<font color="Maroon" title="c6">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="borsuk_1.html#K8" title="BORSUK_1:func.8">Base-Appr</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E7:78_1_1_1"/><span class="lab"><font color="Green" title="E54">A5</font></span>: 
<font color="Maroon" title="c6">K</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:78_1_1_1"/><span class="lab"><font color="Green" title="E55">A6</font></span>: 
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#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T63" target="_self" title="BORSUK_1:th.63">Th63</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">PK</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">Pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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="E10:78_1_1_1"/>
<font color="Maroon" title="c6">K</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78_1_1_1"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="ref" href="borsuk_1.html#T51" target="_self" title="BORSUK_1:th.51">Th51</a>, <a class="ref" href="tops_2.html#T18" title="TOPS_2:th.18">TOPS_2:18</a></span>;<br/><a NAME="E11:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A7</font></span>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">Pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">K</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T60" target="_self" title="BORSUK_1:th.60">Th60</a></span>;<br/><a NAME="E12:78_1_1_1"/>
<font color="Maroon" title="c7">PK</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:78_1_1_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="ref" href="borsuk_1.html#T62" target="_self" title="BORSUK_1:th.62">Th62</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E14:78_1_1_1"/><span class="lab"><font color="Green" title="E57">A8</font></span>: 
<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#K9" title="BORSUK_1:func.9">Pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">K</font>
 <span class="kw">and </span><br/><a NAME="E15:78_1_1_1"/><span class="lab"><font color="Green" title="E58">A9</font></span>: 
<font color="Maroon" title="c8">M</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c4">A</font>
 <span class="kw">and </span><br/><a NAME="E16:78_1_1_1"/><span class="lab"><font color="Green" title="E59">A10</font></span>: 
<font color="Maroon" title="c8">M</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E11:78_1_1_1"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="ref" href="compts_1.html#D7" title="COMPTS_1:def.7">COMPTS_1:def 7</a></span>;<br/><span class="kw">consider </span><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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E18:78_1_1_1"/><span class="lab"><font color="Green" title="E60">A11</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E19:78_1_1_1"/><span class="lab"><font color="Green" title="E61">A12</font></span>: 
<font color="Maroon" title="c9">N</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">and </span><br/><a NAME="E20:78_1_1_1"/><span class="lab"><font color="Green" title="E62">A13</font></span>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">Pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:78_1_1_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="txt" href="#E16:78_1_1_1"><span class="lab"><font color="Green" title="E59">A10</font></span></a>, <a class="ref" href="borsuk_1.html#T64" target="_self" title="BORSUK_1:th.64">Th64</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">F</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">Pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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="E22:78_1_1_1"/><span class="lab"><font color="Green" title="E63">A14</font></span>: 
<font color="Maroon" title="c9">N</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">e1</font>, <font color="Maroon" title="c12">e2</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="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a>,<a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</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="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#K4" title="BORSUK_1:func.4">:]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</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="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">N</font> )</span><br/>

<a NAME="E1:78_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A15</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:78_1_1_1"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:78_1_1_1_1"/><span class="lab"><font color="Green" title="E65">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</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="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#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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</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="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c9">N</font></span><br/>

<a NAME="E3:78_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/>
<a NAME="E4:78_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:78_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1_1_1_1"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></span>;<br/>
<a NAME="E6:78_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78_1_1_1_1"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:78_1_1_1_1"/><span class="lab"><font color="Green" title="E67">A18</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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">X1</font>
 <span class="kw">and </span><br/><a NAME="E9:78_1_1_1_1"/><span class="lab"><font color="Green" title="E68">A19</font></span>: 
<font color="Maroon" title="c13">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1_1_1"><span class="lab"><font color="Green" title="E64">A15</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="c14">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E11:78_1_1_1_1"/><span class="lab"><font color="Green" title="E69">A20</font></span>: 
<font color="Maroon" title="c14">XY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N</font>
 <span class="kw">and </span><br/><a NAME="E12:78_1_1_1_1"/><span class="lab"><font color="Green" title="E70">A21</font></span>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K9" title="BORSUK_1:func.9">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="c14">XY</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:78_1_1_1"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="txt" href="#E9:78_1_1_1_1"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<a NAME="E13:78_1_1_1_1"/>
<font color="Maroon" title="c14">XY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:78_1_1_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="txt" href="#E11:78_1_1_1_1"><span class="lab"><font color="Green" title="E69">A20</font></span></a></span>;<br/>
<a NAME="E14:78_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">XY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K8" title="BORSUK_1:func.8">Base-Appr</a> <font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78_1_1_1"><span class="lab"><font color="Green" title="E53">A4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">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="c16">Y2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E16:78_1_1_1_1"/><span class="lab"><font color="Green" title="E71">A22</font></span>: 
<font color="Maroon" title="c14">XY</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="c15">X2</font>,<font color="Maroon" title="c16">Y2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E17:78_1_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c15">X2</font>,<font color="Maroon" title="c16">Y2</font></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> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E18:78_1_1_1_1"/>
<font color="Maroon" title="c15">X2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E19:78_1_1_1_1"/>
<font color="Maroon" title="c16">Y2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ;<br/>
<a NAME="E20:78_1_1_1_1"/>
<font color="Maroon" title="c14">XY</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78_1_1_1"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="txt" href="#E18:78_1_1_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="txt" href="#E11:78_1_1_1_1"><span class="lab"><font color="Green" title="E69">A20</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">xy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E22:78_1_1_1_1"/><span class="lab"><font color="Green" title="E72">A23</font></span>: 
<font color="Maroon" title="c17">xy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">XY</font>
 <span class="kw">and </span><br/><a NAME="E23:78_1_1_1_1"/><span class="lab"><font color="Green" title="E73">A24</font></span>: 
<font color="Maroon" title="c17">xy</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="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#K4" title="BORSUK_1:func.4">:]</a></span>
 <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="E24:78_1_1_1_1"/>
<font color="Maroon" title="c17">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:78_1_1_1_1"><span class="lab"><font color="Green" title="E73">A24</font></span></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/>
<a NAME="E25:78_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E26:78_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A25</font></span>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:78_1_1_1_1"><span class="lab"><font color="Green" title="E71">A22</font></span></a>, <a class="txt" href="#E22:78_1_1_1_1"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/>
<a NAME="E27:78_1_1_1_1"/>
<font color="Maroon" title="c14">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78_1_1_1_1"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="txt" href="#E12:78_1_1_1_1"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="ref" href="funct_3.html#T9" title="FUNCT_3:th.9">FUNCT_3:9</a></span>;<br/>
<a NAME="E28:78_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c15">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78_1_1_1_1"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="txt" href="#E12:78_1_1_1_1"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="txt" href="#E16:78_1_1_1_1"><span class="lab"><font color="Green" title="E71">A22</font></span></a>, <a class="ref" href="borsuk_1.html#T13" target="_self" title="BORSUK_1:th.13">Th13</a></span>;<br/>
<a NAME="E29:78_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</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="c15">X2</font>,<font color="Maroon" title="c16">Y2</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="#E26:78_1_1_1_1"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="txt" href="#E5:78_1_1_1_1"><span class="lab"><font color="Green" title="E66">A17</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="E30:78_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</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="c9">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:78_1_1_1_1"><span class="lab"><font color="Green" title="E69">A20</font></span></a>, <a class="txt" href="#E16:78_1_1_1_1"><span class="lab"><font color="Green" title="E71">A22</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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="E23:78_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">F</font> is    <a href="setfam_1.html#M1" title="SETFAM_1:mode.1">Cover</a> of <font color="Maroon" title="c4">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T62" target="_self" title="BORSUK_1:th.62">Th62</a></span>;<br/><a NAME="E24:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A26</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">H</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K10" title="BORSUK_1:func.10">Pr2</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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/><div><span class="lab"><font color="Green" title="E65">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">C</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="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">H</font> implies <font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">C</font> )</span><br/><span class="kw">assume </span><a NAME="E1:78_1_1_1_2"/>
<font color="Maroon" title="c12">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">H</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="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">C</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E3:78_1_1_1_2"/><span class="lab"><font color="Green" title="E66">A28</font></span>: 
<font color="Maroon" title="c13">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N</font>
 <span class="kw">and </span><br/><a NAME="E4:78_1_1_1_2"/><span class="lab"><font color="Green" title="E67">A29</font></span>: 
<font color="Maroon" title="c12">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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T58" target="_self" title="BORSUK_1:th.58">Th58</a></span>;<br/><a NAME="E5:78_1_1_1_2"/>
<font color="Maroon" title="c13">D</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78_1_1_1"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="txt" href="#E18:78_1_1_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="txt" href="#E3:78_1_1_1_2"><span class="lab"><font color="Green" title="E66">A28</font></span></a></span>;<br/><a NAME="E6:78_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">D</font> <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="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#K4" title="BORSUK_1:func.4">:]</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> 
 <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>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E8:78_1_1_1_2"/><span class="lab"><font color="Green" title="E68">A30</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">D</font> <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="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#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/><a NAME="E9:78_1_1_1_2"/><span class="lab"><font color="Green" title="E69">A31</font></span>: 
<font color="Maroon" title="c14">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"><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#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78_1_1_1_2"><span class="lab"><font color="Green" title="E68">A30</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="E10:78_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/><a NAME="E11:78_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A32</font></span>: 
<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="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D6" title="FUNCT_3:def.6">FUNCT_3:def 6</a></span>;<br/><a NAME="E12:78_1_1_1_2"/>
<font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:78_1_1_1_2"><span class="lab"><font color="Green" title="E69">A31</font></span></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/><a NAME="E13:78_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E14:78_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A33</font></span>: 
<font color="Maroon" title="c14">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="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></span>;<br/><a NAME="E15:78_1_1_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:78_1_1_1_2"><span class="lab"><font color="Green" title="E68">A30</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E16:78_1_1_1_2"/>
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="txt" href="#E4:78_1_1_1_2"><span class="lab"><font color="Green" title="E67">A29</font></span></a>, <a class="txt" href="#E14:78_1_1_1_2"><span class="lab"><font color="Green" title="E71">A33</font></span></a>, <a class="txt" href="#E11:78_1_1_1_2"><span class="lab"><font color="Green" title="E70">A32</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</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="E27:78_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</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#K4" title="BORSUK_1:func.4">:]</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:78_1_1_1"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="ref" href="setfam_1.html#D12" title="SETFAM_1:def.12">SETFAM_1:def 12</a></span>;<br/><a NAME="E28:78_1_1_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1_1_1"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></span>;<br/><a NAME="E29:78_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T61" target="_self" title="BORSUK_1:th.61">Th61</a></span>;<br/><a NAME="E30:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A34</font></span>: 
<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="c11">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:78_1_1_1"><span class="lab"><font color="Green" title="E65">A27</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/><a NAME="E31:78_1_1_1"/><span class="lab"><font color="Green" title="E67">A35</font></span>: 
<font color="Maroon" title="c9">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="borsuk_1.html#K8" title="BORSUK_1:func.8">Base-Appr</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78_1_1_1"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="txt" href="#E18:78_1_1_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E32:78_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A36</font></span>: 
<font color="Maroon" title="c9">N</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T51" target="_self" title="BORSUK_1:th.51">Th51</a>, <a class="ref" href="tops_2.html#T18" title="TOPS_2:th.18">TOPS_2:18</a></span>;<br/><a NAME="E33:78_1_1_1"/><span class="kw">then </span>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c11">H</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="#E19:78_1_1_1"><span class="lab"><font color="Green" title="E61">A12</font></span></a>, <a class="ref" href="borsuk_1.html#T60" target="_self" title="BORSUK_1:th.60">Th60</a>, <a class="ref" href="tops_2.html#T27" title="TOPS_2:th.27">TOPS_2:27</a></span>;<br/><a NAME="E34:78_1_1_1"/><span class="kw">then </span>
<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="c11">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:78_1_1_1"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">W</font> =  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c11">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></span>;<br/><a NAME="E36:78_1_1_1"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:78_1_1_1"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="ref" href="borsuk_1.html#T60" target="_self" title="BORSUK_1:th.60">Th60</a>, <a class="ref" href="tops_2.html#T26" title="TOPS_2:th.26">TOPS_2:26</a></span>;<br/><a NAME="E37:78_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:78_1_1_1"><span class="lab"><font color="Green" title="E64">A26</font></span></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D2" title="CONNSP_2:def.2">CONNSP_2:def 2</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c14">V</font> = <font color="Maroon" title="c13">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c14">V</font>,<font color="Olive" title="b1">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font></span><br/><span class="kw">take </span><font color="Maroon" title="c15">W</font> = <font color="Maroon" title="c12">W</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#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c14">V</font>,<font color="Maroon" title="c15">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">e</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="c16">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N</font> implies ( <font color="Maroon" title="c16">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="c16">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="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ) )</span><br/><span class="kw">assume </span><a NAME="E1:78_1_1_1_3"/>
<font color="Maroon" title="c16">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">N</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="c16">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="c16">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="b1">X1</font>,<font color="Olive" title="b2">Y1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/><a NAME="E2:78_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c16">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_1.html#K8" title="BORSUK_1:func.8">Base-Appr</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:78_1_1_1"><span class="lab"><font color="Green" title="E67">A35</font></span></a></span>;<br/><a NAME="E3:78_1_1_1_3"/><span class="kw">then </span>
 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="c16">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="b1">X1</font>,<font color="Olive" title="b2">Y1</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"><font color="Olive" title="b1">X1</font>,<font color="Olive" title="b2">Y1</font></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> <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/><span class="kw">hence </span><a NAME="E4:78_1_1_1_3"/>
( <font color="Maroon" title="c16">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="c16">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="b1">X1</font>,<font color="Olive" title="b2">Y1</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"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E40:78_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c14">V</font>,<font color="Maroon" title="c15">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T56" target="_self" title="BORSUK_1:th.56">Th56</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:78_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font></span><br/><div class="add"><a NAME="E2:78_1_1_2"/><span class="kw">then </span>
<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/><span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D2" title="CONNSP_2:def.2">CONNSP_2:def 2</a></span>;<br/><span class="kw">consider </span><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/><span class="kw">take </span><font color="Maroon" title="c8">V</font> = <font color="Maroon" title="c6">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Olive" title="b1">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font></span><br/><span class="kw">take </span><font color="Maroon" title="c9">W</font> = <font color="Maroon" title="c7">W</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#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c9">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font></span><br/><a NAME="E5:78_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c9">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></span>;<br/><span class="kw">hence </span><a NAME="E6:78_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c8">V</font>,<font color="Maroon" title="c9">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:78"/>
 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#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">W</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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