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<div class="add">

<b>let </b><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B1</font> being   <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c1">T1</font><br/> for <font color="Olive" title="b2">B2</font> being   <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c2">T2</font>  st  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b2">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font> holds <br/><span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b3">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>, <font color="Olive" title="b4">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : ( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B1</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B2</font> ) </span>}</span>  is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/><b>let </b><font color="Maroon" title="c3">B1</font> be    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c1">T1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B2</font> being   <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c2">T2</font>  st  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font> holds <br/><span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>, <font color="Olive" title="b3">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : ( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B2</font> ) </span>}</span>  is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/><b>let </b><font color="Maroon" title="c4">B2</font> be    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c2">T2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font> implies <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>, <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> ) </span>}</span>  is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/>





<b>assume </b><a NAME="E1:61"/><i><font color="Green" title="E31">A1</font></i>: 
(  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font> &amp;  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>, <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> ) </span>}</span>  is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/>

<b>set </b><font color="Maroon" title="c5">cT1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>;<br/>
<b>set </b><font color="Maroon" title="c6">cT2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>;<br/>
<b>set </b><font color="Maroon" title="c7">BB1</font> = <span class="p1">{<span class="default"> <span class="p2"><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="c1">T1</font>,<font color="Olive" title="b1">b</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c8">BB2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c9">CC</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>, <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c10">T</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">BB</font> = <span class="p1">{<span class="default"> <span class="p2"><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="c1">T1</font>,<font color="Olive" title="b1">b</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> </span>}</span>  as    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="ref" href="yellow_9.html#T41" target="_self" title="YELLOW_9:th.41">Th41</a></i>;<br/>
<a NAME="E3:61"/><i><font color="Green" title="E32">A2</font></i>: 
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c11">BB</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T23" target="_self" title="YELLOW_9:th.23">Th23</a></i>;<br/>
<a NAME="E4:61"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>, <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>, <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> ) </span>}</span>  or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>, <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/>

<a NAME="E2:61_1"/><b>then </b>
 ex <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="c1">T1</font> ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> )
 
;<br/>
<b>hence </b><a NAME="E3:61_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">CC</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>, <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">CC</font> = <font color="Maroon" title="c12">CC</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ;<br/>
<a NAME="E7:61"/>
<font color="Maroon" title="c13">CC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">CC</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:61_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">CC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c15">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c16">b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font><b> such that </b><br/><a NAME="E3:61_2"/><i><font color="Green" title="E33">A3</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> )
 ;<br/>
<a NAME="E4:61_2"/>
( <font color="Maroon" title="c3">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#D5" title="CANTOR_1:def.5">CANTOR_1:def 5</a></i>;<br/>
<a NAME="E5:61_2"/><b>then </b>
( <font color="Maroon" title="c15">a</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c16">b</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="#E3:61_2"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E6:61_2"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>;<br/>
<b>hence </b><a NAME="E7:61_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="txt" href="#E3:61_2"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:61"/><b>then </b>
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c13">CC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T9" title="CANTOR_1:th.9">CANTOR_1:9</a></i>;<br/>
<a NAME="E9:61"/><b>then </b><i><font color="Green" title="E33">A4</font></i>: 
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c13">CC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T6" title="CANTOR_1:th.6">CANTOR_1:6</a></i>;<br/>
<a NAME="E10:61"/>
<font color="Maroon" title="c11">BB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c13">CC</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">BB</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c13">CC</font> )</span><br/>

<b>assume </b><a NAME="E1:61_3"/><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">BB</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c13">CC</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_3_1"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><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="c1">T1</font>,<font color="Olive" title="b1">b</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> </span>}</span>  or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="#E1:61_3"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_1_1"><b>suppose </b></a><a NAME="E1:61_3_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><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="c1">T1</font>,<font color="Olive" title="b1">b</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c13">CC</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c15">b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font><b> such that </b><br/><a NAME="E3:61_3_1_1"/><i><font color="Green" title="E35">A6</font></i>: 
( <font color="Maroon" title="c14">x</font> <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="c1">T1</font>,<font color="Maroon" title="c15">b</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> )
 ;<br/><b>set </b><font color="Maroon" title="c16">Y</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c15">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> </span>}</span> ;<br/><a NAME="E4:61_3_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c15">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">y</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="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c15">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> </span>}</span>  or <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:61_3_1_1_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c15">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/>

<a NAME="E2:61_3_1_1_1"/><b>then </b>
 ex <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="c1">T1</font> st <br/>( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c15">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> )
 
;<br/>
<b>hence </b><a NAME="E3:61_3_1_1_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c17">Y</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c15">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ;<br/><b>reconsider </b><font color="Maroon" title="c18">Y</font> = <font color="Maroon" title="c17">Y</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ;<br/><a NAME="E7:61_3_1_1"/><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c18">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">CC</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">y</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="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font> or <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">CC</font> )</span><br/>

<b>assume </b><a NAME="E1:61_3_1_1_2"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">CC</font></span><br/>

<a NAME="E2:61_3_1_1_2"/><b>then </b>
 ex <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="c1">T1</font> st <br/>( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c15">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> )
 
;<br/>
<b>hence </b><a NAME="E3:61_3_1_1_2"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">CC</font>
 <b>by </b><i><a class="txt" href="#E3:61_3_1_1"><i><font color="Green" title="E35">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E8:61_3_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c18">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3_1_1_3"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c18">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">x</font></span>
<div class="add"><b>let </b><font color="Maroon" title="c19">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> implies <font color="Maroon" title="c19">z</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="c18">Y</font> )</span><br/><b>assume </b><a NAME="E1:61_3_1_1_3_1"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">z</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="c18">Y</font></span><br/><b>then consider </b><font color="Maroon" title="c20">p1</font>, <font color="Maroon" title="c21">p2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:61_3_1_1_3_1"/><i><font color="Green" title="E37">A8</font></i>: 
( <font color="Maroon" title="c20">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font> &amp; <font color="Maroon" title="c21">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">p1</font>,<font color="Maroon" title="c21">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">z</font> )
 <b>by </b><i><a class="txt" href="#E3:61_3_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:61_3_1_1_3_1"/><i><font color="Green" title="E38">A9</font></i>: 
( <font color="Maroon" title="c20">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">a</font> &amp; <font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> )
 <b>by </b><i><a class="txt" href="#E1:61"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="#E3:61_3_1_1_3_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">a</font> = <font color="Maroon" title="c22">a</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> <b>by </b><i><a class="txt" href="#E5:61_3_1_1_3_1"><i><font color="Green" title="E38">A9</font></i></a></i>;<br/><a NAME="E7:61_3_1_1_3_1"/>
( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c15">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font> &amp; <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c23">a</font>,<font color="Maroon" title="c15">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )
 <b>by </b><i><a class="txt" href="#E3:61_3_1_1_3_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="#E5:61_3_1_1_3_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>hence </b><a NAME="E8:61_3_1_1_3_1"/>
<font color="Maroon" title="c19">z</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="c18">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c19">z</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="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">z</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="c18">Y</font> or <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> )</span><br/>

<b>assume </b><a NAME="E2:61_3_1_1_3"/>
<font color="Maroon" title="c19">z</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="c18">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font></span><br/>

<b>then consider </b><font color="Maroon" title="c20">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:61_3_1_1_3"/><i><font color="Green" title="E37">A10</font></i>: 
( <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">y</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:61_3_1_1_3"/>
 ex <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="c1">T1</font> st <br/>( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Maroon" title="c15">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> )
 
<b>by </b><i><a class="txt" href="#E4:61_3_1_1_3"><i><font color="Green" title="E37">A10</font></i></a></i>;<br/>
<a NAME="E6:61_3_1_1_3"/><b>then </b>
<font color="Maroon" title="c20">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="#E3:61_3_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<b>hence </b><a NAME="E7:61_3_1_1_3"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E4:61_3_1_1_3"><i><font color="Green" title="E37">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:61_3_1_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c13">CC</font>
 <b>by </b><i><a class="txt" href="#E7:61_3_1_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="cantor_1.html#D1" title="CANTOR_1:def.1">CANTOR_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_3_1_2"><b>suppose </b></a><a NAME="E1:61_3_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c13">CC</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c15">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font><b> such that </b><br/><a NAME="E3:61_3_1_2"/><i><font color="Green" title="E35">A11</font></i>: 
( <font color="Maroon" title="c14">x</font> <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"><font color="Maroon" title="c15">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> )
 ;<br/><b>set </b><font color="Maroon" title="c16">Y</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Olive" title="b1">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> </span>}</span> ;<br/><a NAME="E4:61_3_1_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Olive" title="b1">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">y</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="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Olive" title="b1">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> </span>}</span>  or <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:61_3_1_2_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Olive" title="b1">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span></span><br/>

<a NAME="E2:61_3_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> st <br/>( <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Olive" title="b1">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> )
 
;<br/>
<b>hence </b><a NAME="E3:61_3_1_2_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c17">Y</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Olive" title="b1">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ;<br/><b>reconsider </b><font color="Maroon" title="c18">Y</font> = <font color="Maroon" title="c17">Y</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> ;<br/><a NAME="E7:61_3_1_2"/><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c18">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">CC</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">y</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="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font> or <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">CC</font> )</span><br/>

<b>assume </b><a NAME="E1:61_3_1_2_2"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">CC</font></span><br/>

<a NAME="E2:61_3_1_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> st <br/>( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Olive" title="b1">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> )
 
;<br/>
<b>hence </b><a NAME="E3:61_3_1_2_2"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">CC</font>
 <b>by </b><i><a class="txt" href="#E3:61_3_1_2"><i><font color="Green" title="E35">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E8:61_3_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c18">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3_1_2_3"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c18">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">x</font></span>
<div class="add"><b>let </b><font color="Maroon" title="c19">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> implies <font color="Maroon" title="c19">z</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="c18">Y</font> )</span><br/><b>assume </b><a NAME="E1:61_3_1_2_3_1"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">z</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="c18">Y</font></span><br/><b>then consider </b><font color="Maroon" title="c20">p1</font>, <font color="Maroon" title="c21">p2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:61_3_1_2_3_1"/><i><font color="Green" title="E37">A13</font></i>: 
( <font color="Maroon" title="c20">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">a</font> &amp; <font color="Maroon" title="c21">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">p1</font>,<font color="Maroon" title="c21">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">z</font> )
 <b>by </b><i><a class="txt" href="#E3:61_3_1_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c22">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:61_3_1_2_3_1"/><i><font color="Green" title="E38">A14</font></i>: 
( <font color="Maroon" title="c21">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">b</font> &amp; <font color="Maroon" title="c22">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> )
 <b>by </b><i><a class="txt" href="#E1:61"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="#E3:61_3_1_2_3_1"><i><font color="Green" title="E37">A13</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">b</font> = <font color="Maroon" title="c22">b</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> <b>by </b><i><a class="txt" href="#E5:61_3_1_2_3_1"><i><font color="Green" title="E38">A14</font></i></a></i>;<br/><a NAME="E7:61_3_1_2_3_1"/>
( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c23">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font> &amp; <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c23">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> )
 <b>by </b><i><a class="txt" href="#E3:61_3_1_2_3_1"><i><font color="Green" title="E37">A13</font></i></a>, <a class="txt" href="#E5:61_3_1_2_3_1"><i><font color="Green" title="E38">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>hence </b><a NAME="E8:61_3_1_2_3_1"/>
<font color="Maroon" title="c19">z</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="c18">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c19">z</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="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c19">z</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="c18">Y</font> or <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font> )</span><br/>

<b>assume </b><a NAME="E2:61_3_1_2_3"/>
<font color="Maroon" title="c19">z</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="c18">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font></span><br/>

<b>then consider </b><font color="Maroon" title="c20">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:61_3_1_2_3"/><i><font color="Green" title="E37">A15</font></i>: 
( <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">y</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:61_3_1_2_3"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> st <br/>( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Olive" title="b1">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> )
 
<b>by </b><i><a class="txt" href="#E4:61_3_1_2_3"><i><font color="Green" title="E37">A15</font></i></a></i>;<br/>
<a NAME="E6:61_3_1_2_3"/><b>then </b>
<font color="Maroon" title="c20">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">x</font>
 
<b>by </b><i><a class="txt" href="#E3:61_3_1_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></i>;<br/>
<b>hence </b><a NAME="E7:61_3_1_2_3"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E4:61_3_1_2_3"><i><font color="Green" title="E37">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:61_3_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c13">CC</font>
 <b>by </b><i><a class="txt" href="#E7:61_3_1_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="cantor_1.html#D1" title="CANTOR_1:def.1">CANTOR_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:61"/><b>then </b>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c11">BB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c13">CC</font></span>)</span>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T16" title="CANTOR_1:th.16">CANTOR_1:16</a></i>;<br/>
<a NAME="E12:61"/><b>then </b>
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c13">CC</font> is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:61"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="#E9:61"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="cantor_1.html#D5" title="CANTOR_1:def.5">CANTOR_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:61"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font>, <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B2</font> ) </span>}</span>  is    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">T1</font>,<font color="Maroon" title="c2">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_9.html#T24" target="_self" title="YELLOW_9:th.24">Th24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
