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<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c4">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">R</font> being   <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</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">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</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">S2</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</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="c4">T2</font> holds <br/><span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U1</font>,<font color="Olive" title="b4">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b3">U2</font>,<font color="Olive" title="b5">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b2">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b3">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b4">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b5">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b2">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b5">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span>  is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Olive" title="b1">R</font></span><br/><b>let </b><font color="Maroon" title="c5">R</font> be    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</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">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">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"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</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">S2</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</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="c4">T2</font> implies <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span>  is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c5">R</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green" title="E26">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</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">S2</font>
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green" title="E27">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</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="c4">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">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span>  is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c5">R</font></span><br/>

<b>set </b><font color="Maroon" title="c6">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ;<br/>
<a NAME="E3:60"/><i><font color="Green" title="E28">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="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">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E4:60"/><i><font color="Green" title="E29">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="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">S2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">BST</font> =  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</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">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,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="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> as    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c5">R</font> <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="#E3:60"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="yellow_9.html#T60" title="YELLOW_9:th.60">YELLOW_9:60</a></i>;<br/>
<a NAME="E6:60"/><i><font color="Green" title="E30">A5</font></i>: 
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">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <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">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b2">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b3">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font> : ( <font color="Olive" title="b2">X1</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 <font color="Maroon" title="c1">S1</font> &amp; <font color="Olive" title="b3">Y1</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 <font color="Maroon" title="c3">T1</font> ) </span>}</span>  </span>}</span> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E7:60"/><i><font color="Green" title="E31">A6</font></i>: 
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="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <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="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> : <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b2">X1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">Y1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b2">X1</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 <font color="Maroon" title="c2">S2</font> &amp; <font color="Olive" title="b3">Y1</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 <font color="Maroon" title="c4">T2</font> ) </span>}</span>  </span>}</span> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<i><font color="Green" title="E32">A7</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">R</font> = 
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">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="c2">S2</font>,<font color="Maroon" title="c4">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#D6" title="YELLOW_9:def.6">YELLOW_9:def 6</a></i>
<br/>.= 
<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">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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">S2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E3:60"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>
<br/>.= 
<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">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E27">A2</font></i></a></i>
;<br/>
<a NAME="E9:60"/><i><font color="Green" title="E33">A8</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span>  <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="c5">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">c</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="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span>  or <font color="Maroon" title="c8">c</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 <font color="Maroon" title="c5">R</font> )</span><br/>

<b>assume </b><a NAME="E1:60_2"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </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="c8">c</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 <font color="Maroon" title="c5">R</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">U1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c10">U2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c11">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c12">V2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font><b> such that </b><br/><a NAME="E3:60_2"/><i><font color="Green" title="E34">A9</font></i>: 
<font color="Maroon" title="c8">c</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="c9">U1</font>,<font color="Maroon" title="c11">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c10">U2</font>,<font color="Maroon" title="c12">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E4:60_2"/><i><font color="Green" title="E35">A10</font></i>: 
( <font color="Maroon" title="c9">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c11">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/>
<a NAME="E5:60_2"/>
( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c9">U1</font>,<font color="Maroon" title="c11">V1</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> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c10">U2</font>,<font color="Maroon" title="c12">V2</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="txt" href="#E4:60_2"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>;<br/>
<a NAME="E6:60_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="c9">U1</font>,<font color="Maroon" title="c11">V1</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> 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">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c10">U2</font>,<font color="Maroon" title="c12">V2</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> 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="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:60_2"/><b>then </b><i><font color="Green" title="E36">A11</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">BST</font>
 
<b>by </b><i><a class="txt" href="#E3:60_2"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E8:60_2"/>
<font color="Maroon" title="c7">BST</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="c5">R</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2" title="CANTOR_1:def.2">CANTOR_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E9:60_2"/>
<font color="Maroon" title="c8">c</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 <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="#E7:60_2"><i><font color="Green" title="E36">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="E10:60"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </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 <font color="Maroon" title="c5">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">c</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="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span>  or <font color="Maroon" title="c8">c</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 <font color="Maroon" title="c5">R</font> )</span><br/>

<b>assume </b><a NAME="E1:60_3"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </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="c8">c</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 <font color="Maroon" title="c5">R</font></span><br/>

<b>then consider </b><font color="Maroon" title="c9">U1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c10">U2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c11">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c12">V2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font><b> such that </b><br/><a NAME="E3:60_3"/><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c8">c</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="c9">U1</font>,<font color="Maroon" title="c11">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c10">U2</font>,<font color="Maroon" title="c12">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E4:60_3"/>
( <font color="Maroon" title="c9">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c10">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c11">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c12">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/>
<b>thus </b><a NAME="E5:60_3"/>
<font color="Maroon" title="c8">c</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 <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="#E4:60"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="#E8:60"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="#E3:60_3"><i><font color="Green" title="E34">A12</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>then reconsider </b><font color="Maroon" title="c8">C1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c5">R</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">C1</font> = <font color="Maroon" title="c8">C1</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c5">R</font> ;<br/>
<a NAME="E13:60"/>
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="c5">R</font>  st <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/>for <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">R</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/> ex <font color="Olive" title="b3">a</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">R</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C1</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">R</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> holds <br/> ex <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="c5">R</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C1</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> ) )</span><br/>

<b>assume </b><a NAME="E1:60_4"/><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c10">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">R</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> holds <br/> ex <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="c5">R</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C1</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b2">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> )</span><br/>

<b>let </b><font color="Maroon" title="c11">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font> implies  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="c5">R</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C1</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> ) )</span><br/>

<b>assume </b><a NAME="E2:60_4"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c5">R</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C1</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c12">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">R</font><b> such that </b><br/><a NAME="E4:60_4"/><i><font color="Green" title="E35">A14</font></i>: 
( <font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">BST</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font> &amp; <font color="Maroon" title="c12">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> )
 <b>by </b><i><a class="txt" href="#E1:60_4"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="yellow_9.html#T31" title="YELLOW_9:th.31">YELLOW_9:31</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">X1</font>, <font color="Maroon" title="c14">X2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:60_4"/><i><font color="Green" title="E36">A15</font></i>: 
<font color="Maroon" title="c13">X1</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">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>and </b><br/><a NAME="E7:60_4"/><i><font color="Green" title="E37">A16</font></i>: 
<font color="Maroon" title="c14">X2</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="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <b>and </b><br/><a NAME="E8:60_4"/><i><font color="Green" title="E38">A17</font></i>: 
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">X1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c14">X2</font>
 <b>by </b><i><a class="txt" href="#E4:60_4"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">F1</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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E10:60_4"/><i><font color="Green" title="E39">A18</font></i>: 
<font color="Maroon" title="c13">X1</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="c15">F1</font>
 <b>and </b><br/><a NAME="E11:60_4"/><i><font color="Green" title="E40">A19</font></i>: 
<font color="Maroon" title="c15">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">K1</font>,<font color="Olive" title="b2">L1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">K1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">L1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font> : ( <font color="Olive" title="b1">K1</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 <font color="Maroon" title="c1">S1</font> &amp; <font color="Olive" title="b2">L1</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 <font color="Maroon" title="c3">T1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E6:60"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="#E6:60_4"><i><font color="Green" title="E36">A15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">F2</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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><b> such that </b><br/><a NAME="E13:60_4"/><i><font color="Green" title="E41">A20</font></i>: 
<font color="Maroon" title="c14">X2</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="c16">F2</font>
 <b>and </b><br/><a NAME="E14:60_4"/><i><font color="Green" title="E42">A21</font></i>: 
<font color="Maroon" title="c16">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">K2</font>,<font color="Olive" title="b2">L2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">K2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b2">L2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">K2</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 <font color="Maroon" title="c2">S2</font> &amp; <font color="Olive" title="b2">L2</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 <font color="Maroon" title="c4">T2</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E7:60"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="#E7:60_4"><i><font color="Green" title="E37">A16</font></i></a></i>;<br/>
<a NAME="E15:60_4"/><i><font color="Green" title="E43">A22</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">X1</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">X2</font> )
 
<b>by </b><i><a class="txt" href="#E4:60_4"><i><font color="Green" title="E35">A14</font></i></a>, <a class="txt" href="#E8:60_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">G1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:60_4"/><i><font color="Green" title="E44">A23</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c17">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">F1</font> )
 <b>by </b><i><a class="txt" href="#E10:60_4"><i><font color="Green" title="E39">A18</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">G2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E19:60_4"/><i><font color="Green" title="E45">A24</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">G2</font> &amp; <font color="Maroon" title="c18">G2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">F2</font> )
 <b>by </b><i><a class="txt" href="#E13:60_4"><i><font color="Green" title="E41">A20</font></i></a>, <a class="txt" href="#E15:60_4"><i><font color="Green" title="E43">A22</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E20:60_4"/>
<font color="Maroon" title="c17">G1</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">K1</font>,<font color="Olive" title="b2">L1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">K1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">L1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font> : ( <font color="Olive" title="b1">K1</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 <font color="Maroon" title="c1">S1</font> &amp; <font color="Olive" title="b2">L1</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 <font color="Maroon" title="c3">T1</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E11:60_4"><i><font color="Green" title="E40">A19</font></i></a>, <a class="txt" href="#E17:60_4"><i><font color="Green" title="E44">A23</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">K1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c20">L1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font><b> such that </b><br/><a NAME="E22:60_4"/><i><font color="Green" title="E46">A25</font></i>: 
<font color="Maroon" title="c17">G1</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="c19">K1</font>,<font color="Maroon" title="c20">L1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E23:60_4"/><i><font color="Green" title="E47">A26</font></i>: 
( <font color="Maroon" title="c19">K1</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 <font color="Maroon" title="c1">S1</font> &amp; <font color="Maroon" title="c20">L1</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 <font color="Maroon" title="c3">T1</font> )
 ;<br/>
<a NAME="E24:60_4"/>
<font color="Maroon" title="c18">G2</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">K2</font>,<font color="Olive" title="b2">L2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> where <font color="Olive" title="b1">K2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b2">L2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">K2</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 <font color="Maroon" title="c2">S2</font> &amp; <font color="Olive" title="b2">L2</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 <font color="Maroon" title="c4">T2</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E14:60_4"><i><font color="Green" title="E42">A21</font></i></a>, <a class="txt" href="#E19:60_4"><i><font color="Green" title="E45">A24</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">K2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c22">L2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font><b> such that </b><br/><a NAME="E26:60_4"/><i><font color="Green" title="E48">A27</font></i>: 
<font color="Maroon" title="c18">G2</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="c21">K2</font>,<font color="Maroon" title="c22">L2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>and </b><br/><a NAME="E27:60_4"/><i><font color="Green" title="E49">A28</font></i>: 
( <font color="Maroon" title="c21">K2</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 <font color="Maroon" title="c2">S2</font> &amp; <font color="Maroon" title="c22">L2</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 <font color="Maroon" title="c4">T2</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c23">a</font> = <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">K1</font>,<font color="Maroon" title="c20">L1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">K2</font>,<font color="Maroon" title="c22">L2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">R</font> <b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="#E8:60"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c23">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C1</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">a</font> &amp; <font color="Maroon" title="c23">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> )</span><br/>

<a NAME="E29:60_4"/>
( <font color="Maroon" title="c19">K1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c21">K2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c20">L1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c22">L2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="#E23:60_4"><i><font color="Green" title="E47">A26</font></i></a>, <a class="txt" href="#E27:60_4"><i><font color="Green" title="E49">A28</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E30:60_4"/>
<font color="Maroon" title="c23">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">a</font> &amp; <font color="Maroon" title="c23">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> )</span><br/>

<b>thus </b><a NAME="E31:60_4"/>
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">a</font>
 <b>by </b><i><a class="txt" href="#E17:60_4"><i><font color="Green" title="E44">A23</font></i></a>, <a class="txt" href="#E19:60_4"><i><font color="Green" title="E45">A24</font></i></a>, <a class="txt" href="#E22:60_4"><i><font color="Green" title="E46">A25</font></i></a>, <a class="txt" href="#E26:60_4"><i><font color="Green" title="E48">A27</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c23">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font></span><br/>

<a NAME="E32:60_4"/>
( <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c19">K1</font>,<font color="Maroon" title="c20">L1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">X1</font> &amp; <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c21">K2</font>,<font color="Maroon" title="c22">L2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">X2</font> )
 
<b>by </b><i><a class="txt" href="#E10:60_4"><i><font color="Green" title="E39">A18</font></i></a>, <a class="txt" href="#E13:60_4"><i><font color="Green" title="E41">A20</font></i></a>, <a class="txt" href="#E17:60_4"><i><font color="Green" title="E44">A23</font></i></a>, <a class="txt" href="#E19:60_4"><i><font color="Green" title="E45">A24</font></i></a>, <a class="txt" href="#E22:60_4"><i><font color="Green" title="E46">A25</font></i></a>, <a class="txt" href="#E26:60_4"><i><font color="Green" title="E48">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E33:60_4"/><b>then </b>
<font color="Maroon" title="c23">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">X</font>
 
<b>by </b><i><a class="txt" href="#E8:60_4"><i><font color="Green" title="E38">A17</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<b>hence </b><a NAME="E34:60_4"/>
<font color="Maroon" title="c23">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font>
 <b>by </b><i><a class="txt" href="#E4:60_4"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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>
<b>hence </b><a NAME="E14:60"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span></span>)</span> where <font color="Olive" title="b1">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>, <font color="Olive" title="b2">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>, <font color="Olive" title="b3">V1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font>, <font color="Olive" title="b4">V2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> : ( <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">U2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">V1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b4">V2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span>  is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c5">R</font>
 <b>by </b><i><a class="txt" href="#E9:60"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="yellow_9.html#T32" title="YELLOW_9:th.32">YELLOW_9:32</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>
