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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U1</font>,<font color="Olive" title="b4">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b3">U2</font>,<font color="Olive" title="b5">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Olive" title="b1">R</font></span><br/><span class="kw">let </span><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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c5">R</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:60"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
( 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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c5">R</font></span><br/>

<span class="kw">set </span><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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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="E2:60"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<a NAME="E3:60"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> as   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c5">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="yellow_9.html#T60" title="YELLOW_9:th.60">YELLOW_9:60</a></span>;<br/>
<span class="lab"><font color="Green" title="E29">A4</font></span>: 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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D6" title="YELLOW_9:def.6">YELLOW_9:def 6</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>
;<br/>
<a NAME="E6:60"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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="setfam_1.html#K9" title="SETFAM_1:func.9">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_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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="setfam_1.html#K9" title="SETFAM_1:func.9">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/>

<span class="kw">assume </span><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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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/>

<a NAME="E2:60_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">U1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font> ex <font color="Olive" title="b2">U2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font> ex <font color="Olive" title="b3">V1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font> ex <font color="Olive" title="b4">V2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font> st <br/>( <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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> &amp; <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>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:60_2"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E5:60"><span class="lab"><font color="Green" title="E29">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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/>
<span class="kw">reconsider </span><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="E9:60"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<a NAME="E10:60"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">X1</font>,<font color="Olive" title="b3">Y1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<a NAME="E11:60"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
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_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:60_3"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E2:60_3"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:60_3"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:60_3"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:60_3"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_3"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="yellow_9.html#T31" title="YELLOW_9:th.31">YELLOW_9:31</a></span>;<br/>
<span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E8:60_3"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:60_3"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:60_3"/><span class="lab"><font color="Green" title="E39">A14</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_3"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></span>;<br/>
<span class="kw">consider </span><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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E12:60_3"/><span class="lab"><font color="Green" title="E40">A15</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E13:60_3"/><span class="lab"><font color="Green" title="E41">A16</font></span>: 
<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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">K1</font>,<font color="Olive" title="b2">L1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E8:60_3"><span class="lab"><font color="Green" title="E37">A12</font></span></a></span>;<br/>
<a NAME="E14:60_3"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_3"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E10:60_3"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">G1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:60_3"/><span class="lab"><font color="Green" title="E42">A17</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">G1</font>
 <span class="kw">and </span><br/><a NAME="E17:60_3"/><span class="lab"><font color="Green" title="E43">A18</font></span>: 
<font color="Maroon" title="c16">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:60_3"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E18:60_3"/><span class="lab"><font color="Green" title="E44">A19</font></span>: 
<font color="Maroon" title="c16">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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">K1</font>,<font color="Olive" title="b2">L1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:60_3"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="txt" href="#E17:60_3"><span class="lab"><font color="Green" title="E43">A18</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span><span class="kw"> such that </span><br/><a NAME="E20:60_3"/><span class="lab"><font color="Green" title="E45">A20</font></span>: 
<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="c17">F2</font>
 <span class="kw">and </span><br/><a NAME="E21:60_3"/><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<font color="Maroon" title="c17">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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">K2</font>,<font color="Olive" title="b2">L2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:60"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E9:60_3"><span class="lab"><font color="Green" title="E38">A13</font></span></a></span>;<br/>
<a NAME="E22:60_3"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_3"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E10:60_3"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">G2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E24:60_3"/><span class="lab"><font color="Green" title="E47">A22</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E25:60_3"/><span class="lab"><font color="Green" title="E48">A23</font></span>: 
<font color="Maroon" title="c18">G2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:60_3"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E26:60_3"/>
<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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">K2</font>,<font color="Olive" title="b2">L2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:60_3"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="txt" href="#E25:60_3"><span class="lab"><font color="Green" title="E48">A23</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">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="c20">L2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font><span class="kw"> such that </span><br/><a NAME="E28:60_3"/><span class="lab"><font color="Green" title="E49">A24</font></span>: 
<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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c19">K2</font>,<font color="Maroon" title="c20">L2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E29:60_3"/><span class="lab"><font color="Green" title="E50">A25</font></span>: 
( <font color="Maroon" title="c19">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="c20">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/>
<a NAME="E30:60_3"/><span class="lab"><font color="Green" title="E51">A26</font></span>: 
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c19">K2</font>,<font color="Maroon" title="c20">L2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">X2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:60_3"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="txt" href="#E25:60_3"><span class="lab"><font color="Green" title="E48">A23</font></span></a>, <a class="txt" href="#E28:60_3"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E31:60_3"/><span class="lab"><font color="Green" title="E52">A27</font></span>: 
( <font color="Maroon" title="c19">K2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c20">L2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:60_3"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">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="c22">L1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T1</font><span class="kw"> such that </span><br/><a NAME="E33:60_3"/><span class="lab"><font color="Green" title="E53">A28</font></span>: 
<font color="Maroon" title="c16">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c21">K1</font>,<font color="Maroon" title="c22">L1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E34:60_3"/><span class="lab"><font color="Green" title="E54">A29</font></span>: 
( <font color="Maroon" title="c21">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="c22">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E18:60_3"><span class="lab"><font color="Green" title="E44">A19</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">a</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c21">K1</font>,<font color="Maroon" title="c22">L1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c19">K2</font>,<font color="Maroon" title="c20">L2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E5:60"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c23">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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="E36:60_3"/>
( <font color="Maroon" title="c21">K1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c22">L1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:60_3"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E37:60_3"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:60_3"><span class="lab"><font color="Green" title="E52">A27</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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/>

<span class="kw">thus </span><a NAME="E38:60_3"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:60_3"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="txt" href="#E24:60_3"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="txt" href="#E33:60_3"><span class="lab"><font color="Green" title="E53">A28</font></span></a>, <a class="txt" href="#E28:60_3"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="E39:60_3"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c21">K1</font>,<font color="Maroon" title="c22">L1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">X1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:60_3"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="txt" href="#E17:60_3"><span class="lab"><font color="Green" title="E43">A18</font></span></a>, <a class="txt" href="#E33:60_3"><span class="lab"><font color="Green" title="E53">A28</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E40:60_3"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:60_3"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="txt" href="#E30:60_3"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E41:60_3"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_3"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:60"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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="c10">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/>

<a NAME="E1:60_4"/><span class="lab"><font color="Green" title="E33">A30</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#T78" title="TOPS_2:th.78">TOPS_2:78</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:60_4"/>
<font color="Maroon" title="c10">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#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">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="c12">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="c13">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="c14">V2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">T2</font><span class="kw"> such that </span><br/><a NAME="E4:60_4"/><span class="lab"><font color="Green" title="E34">A31</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c11">U1</font>,<font color="Maroon" title="c13">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c12">U2</font>,<font color="Maroon" title="c14">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:60_4"/><span class="lab"><font color="Green" title="E35">A32</font></span>: 
<font color="Maroon" title="c11">U1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E6:60_4"/><span class="lab"><font color="Green" title="E36">A33</font></span>: 
<font color="Maroon" title="c12">U2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E7:60_4"/><span class="lab"><font color="Green" title="E37">A34</font></span>: 
<font color="Maroon" title="c13">V1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E8:60_4"/><span class="lab"><font color="Green" title="E38">A35</font></span>: 
<font color="Maroon" title="c14">V2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ;<br/>
<a NAME="E9:60_4"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c12">U2</font>,<font color="Maroon" title="c14">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_4"><span class="lab"><font color="Green" title="E36">A33</font></span></a>, <a class="txt" href="#E8:60_4"><span class="lab"><font color="Green" title="E38">A35</font></span></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>;<br/>
<a NAME="E10:60_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A36</font></span>: 
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c12">U2</font>,<font color="Maroon" title="c14">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E11:60_4"/>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c11">U1</font>,<font color="Maroon" title="c13">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:60_4"><span class="lab"><font color="Green" title="E35">A32</font></span></a>, <a class="txt" href="#E7:60_4"><span class="lab"><font color="Green" title="E37">A34</font></span></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>;<br/>
<a NAME="E12:60_4"/><span class="kw">then </span>
<span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Maroon" title="c11">U1</font>,<font color="Maroon" title="c13">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E13:60_4"/><span class="kw">then </span>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">BST</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_4"><span class="lab"><font color="Green" title="E34">A31</font></span></a>, <a class="txt" href="#E10:60_4"><span class="lab"><font color="Green" title="E39">A36</font></span></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:60_4"/>
<font color="Maroon" title="c10">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_4"><span class="lab"><font color="Green" title="E33">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:60"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b1">U1</font>,<font color="Olive" title="b3">V1</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><font color="Olive" title="b2">U2</font>,<font color="Olive" title="b4">V2</font></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</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#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c5">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:60"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="yellow_9.html#T32" title="YELLOW_9:th.32">YELLOW_9:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
