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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">S9</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">T9</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">T</font> holds   <a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Olive" title="b1">S9</font>,<font color="Olive" title="b2">T9</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">ST</font> being   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">BX</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">ST</font> : verum </span>}</span>  as   <a href="cantor_1.html#NM2" title="CANTOR_1:NM.2">prebasis</a> of <font color="Maroon" title="c3">ST</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel19.html#D1" target="_self" title="WAYBEL19:def.1">Def1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">S9</font> be   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">T9</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">T</font> holds   <a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">S9</font>,<font color="Olive" title="b1">T9</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">BS</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S9</font> : verum </span>}</span>  as   <a href="cantor_1.html#NM2" title="CANTOR_1:NM.2">prebasis</a> of <font color="Maroon" title="c5">S9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel19.html#D1" target="_self" title="WAYBEL19:def.1">Def1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">T9</font> be   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">S9</font>,<font color="Maroon" title="c7">T9</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">SxT</font> = <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c5">S9</font>,<font color="Maroon" title="c7">T9</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">B2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">T9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">S9</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">BS</font> </span>}</span> ;<br/>
<a NAME="E4:24"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">T9</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c7">T9</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">T</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">BT</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">T9</font> : verum </span>}</span>  as   <a href="cantor_1.html#NM2" title="CANTOR_1:NM.2">prebasis</a> of <font color="Maroon" title="c7">T9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel19.html#D1" target="_self" title="WAYBEL19:def.1">Def1</a></span>;<br/>
<a NAME="E6:24"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S9</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">S9</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E7:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">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="c5">S9</font>,<font color="Maroon" title="c7">T9</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">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</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="#E4:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></span>;<br/>
<a NAME="E8:24"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">ST</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">ST</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E9:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A5</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">ST</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</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="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></span>;<br/>
<a NAME="E10:24"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c4">BX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c5">S9</font>,<font color="Maroon" title="c7">T9</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">z</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="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BX</font> or <font color="Maroon" title="c11">z</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="c5">S9</font>,<font color="Maroon" title="c7">T9</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> )</span><br/>

<a NAME="E1:24_1"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">T9</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="ref" href="tops_1.html#T53" title="TOPS_1:th.53">TOPS_1:53</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:24_1"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BX</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">z</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="c5">S9</font>,<font color="Maroon" title="c7">T9</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span></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#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">ST</font><span class="kw"> such that </span><br/><a NAME="E4:24_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">s</font>, <font color="Maroon" title="c14">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:24_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 <span class="kw">and </span><br/><a NAME="E7:24_1"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>
 <span class="kw">and </span><br/><a NAME="E8:24_1"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c14">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">t</font> = <font color="Maroon" title="c14">t</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">s</font> = <font color="Maroon" title="c13">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>;<br/>
<a NAME="E11:24_1"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c16">s</font>,<font color="Maroon" title="c15">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>;<br/>
<a NAME="E12:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K13" title="YELLOW_3:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c16">s</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K13" title="YELLOW_3:func.13">:]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="yellow_3.html#K13" title="YELLOW_3:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c15">t</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span><a href="yellow_3.html#K13" title="YELLOW_3:func.13">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E4:24_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="waybel19.html#T14" target="_self" title="WAYBEL19:th.14">Th14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">s9</font> = <font color="Maroon" title="c16">s</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">t9</font> = <font color="Maroon" title="c15">t</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">T9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>;<br/>
<a NAME="E15:24_1"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c18">t9</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">BT</font>
 
;<br/>
<a NAME="E16:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c18">t9</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">A1</font> = <span class="p1"><a href="yellow_3.html#K13" title="YELLOW_3:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c16">s</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K13" title="YELLOW_3:func.13">:]</a></span>, <font color="Maroon" title="c20">A2</font> = <span class="p1"><a href="yellow_3.html#K13" title="YELLOW_3:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c15">t</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span><a href="yellow_3.html#K13" title="YELLOW_3:func.13">:]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c5">S9</font>,<font color="Maroon" title="c7">T9</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="txt" href="#E7:24"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></span>;<br/>
<a NAME="E18:24_1"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c5">S9</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="ref" href="tops_1.html#T53" title="TOPS_1:th.53">TOPS_1:53</a></span>;<br/>
<a NAME="E19:24_1"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">s9</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">BS</font>
 
;<br/>
<a NAME="E20:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">s9</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></span>;<br/>
<a NAME="E21:24_1"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c18">t9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>;<br/>
<a NAME="E22:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c20">A2</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="#E16:24_1"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E18:24_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>;<br/>
<a NAME="E23:24_1"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c16">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">s9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>;<br/>
<a NAME="E24:24_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">A1</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="#E20:24_1"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></span>;<br/>
<a NAME="E25:24_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c20">A2</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="#E22:24_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="tops_1.html#T37" title="TOPS_1:th.37">TOPS_1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:24_1"/>
<font color="Maroon" title="c11">z</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="c5">S9</font>,<font color="Maroon" title="c7">T9</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="txt" href="#E12:24_1"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</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">set </span><font color="Maroon" title="c11">B1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S9</font>,<font color="Olive" title="b1">b</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T9</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">BT</font> </span>}</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">BB</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S9</font>,<font color="Olive" title="b1">b</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T9</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">BT</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">T9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">S9</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">BS</font> </span>}</span>  as   <a href="cantor_1.html#NM2" title="CANTOR_1:NM.2">prebasis</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c5">S9</font>,<font color="Maroon" title="c7">T9</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#T41" title="YELLOW_9:th.41">YELLOW_9:41</a></span>;<br/>
<a NAME="E12:24"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c5">S9</font>,<font color="Maroon" title="c7">T9</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> 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="c5">S9</font>,<font color="Maroon" title="c7">T9</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="cantor_1.html#T6" title="CANTOR_1:th.6">CANTOR_1:6</a></span>;<br/>
<a NAME="E13:24"/>
<font color="Maroon" title="c12">BB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">BX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">z</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="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">BB</font> or <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BX</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_2"/><span class="lab"><font color="Green" title="E24">A20</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">BB</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="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BX</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_2_1"/>
( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S9</font>,<font color="Olive" title="b1">b</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T9</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">BT</font> </span>}</span>  or <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">T9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">S9</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">BS</font> </span>}</span>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_2"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_2_1_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S9</font>,<font color="Olive" title="b1">b</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T9</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">BT</font> </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="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BX</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">b</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T9</font><span class="kw"> such that </span><br/><a NAME="E3:24_2_1_1"/><span class="lab"><font color="Green" title="E25">A21</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S9</font>,<font color="Maroon" title="c14">b</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:24_2_1_1"/><span class="lab"><font color="Green" title="E26">A22</font></span>: 
<font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">BT</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c15">t9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">T9</font><span class="kw"> such that </span><br/><a NAME="E6:24_2_1_1"/><span class="lab"><font color="Green" title="E27">A23</font></span>: 
<font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c15">t9</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_2_1_1"><span class="lab"><font color="Green" title="E26">A22</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">t</font> = <font color="Maroon" title="c15">t9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c16">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">ST</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E20">A4</font></span></a></span>;<br/><a NAME="E9:24_2_1_1"/><span class="lab"><font color="Green" title="E28">A24</font></span>: 
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c16">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>;<br/><a NAME="E10:24_2_1_1"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel14.html#T10" title="WAYBEL14:th.10">WAYBEL14:10</a></span>;<br/><a NAME="E11:24_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A25</font></span>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/><a NAME="E12:24_2_1_1"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c15">t9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p3">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">S</font></span>)</span>,<font color="Maroon" title="c16">t</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</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="c1">S</font>,<font color="Maroon" title="c14">b</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="#E11:24_2_1_1"><span class="lab"><font color="Green" title="E29">A25</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E6:24_2_1_1"><span class="lab"><font color="Green" title="E27">A23</font></span></a>, <a class="ref" href="waybel19.html#T14" target="_self" title="WAYBEL19:th.14">Th14</a></span>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <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="c1">S</font>,<font color="Maroon" title="c14">b</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="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></span>
<br/>.= 
<font color="Maroon" title="c13">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:24_2_1_1"><span class="lab"><font color="Green" title="E25">A21</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E14:24_2_1_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E9:24_2_1_1"><span class="lab"><font color="Green" title="E28">A24</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_2_1_2"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">T9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">S9</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">BS</font> </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="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BX</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">a</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">S9</font><span class="kw"> such that </span><br/><a NAME="E3:24_2_1_2"/><span class="lab"><font color="Green" title="E25">A26</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c14">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">T9</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:24_2_1_2"/><span class="lab"><font color="Green" title="E26">A27</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">BS</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c15">s9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">S9</font><span class="kw"> such that </span><br/><a NAME="E6:24_2_1_2"/><span class="lab"><font color="Green" title="E27">A28</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c15">s9</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_2_1_2"><span class="lab"><font color="Green" title="E26">A27</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">s</font> = <font color="Maroon" title="c15">s9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E18">A2</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">x</font> = <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">ST</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E20">A4</font></span></a></span>;<br/><a NAME="E9:24_2_1_2"/><span class="lab"><font color="Green" title="E28">A29</font></span>: 
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c16">s</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>;<br/><a NAME="E10:24_2_1_2"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span> <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">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel14.html#T10" title="WAYBEL14:th.10">WAYBEL14:10</a></span>;<br/><a NAME="E11:24_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A30</font></span>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/><a NAME="E12:24_2_1_2"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c16">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c15">s9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c16">s</font>,<span class="p3">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c2">T</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  = 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c14">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</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="c1">S</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </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="#E11:24_2_1_2"><span class="lab"><font color="Green" title="E29">A30</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E6:24_2_1_2"><span class="lab"><font color="Green" title="E27">A28</font></span></a>, <a class="ref" href="waybel19.html#T14" target="_self" title="WAYBEL19:th.14">Th14</a></span>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c14">a</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</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> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></span>
<br/>.= 
<font color="Maroon" title="c13">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E3:24_2_1_2"><span class="lab"><font color="Green" title="E25">A26</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E14:24_2_1_2"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">BX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E9:24_2_1_2"><span class="lab"><font color="Green" title="E28">A29</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E14:24"/><span class="kw">then </span>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c12">BB</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c4">BX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="cantor_1.html#T16" title="CANTOR_1:th.16">CANTOR_1:16</a></span>;<br/>
<a NAME="E15:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A31</font></span>: 
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c12">BB</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c4">BX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="cantor_1.html#T9" title="CANTOR_1:th.9">CANTOR_1:9</a></span>;<br/>
<a NAME="E16:24"/>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c12">BB</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c5">S9</font>,<font color="Maroon" title="c7">T9</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#T23" title="YELLOW_9:th.23">YELLOW_9:23</a></span>;<br/>
<a NAME="E17:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A32</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="c5">S9</font>,<font color="Maroon" title="c7">T9</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>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c12">BB</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#T22" title="YELLOW_9:th.22">YELLOW_9:22</a></span>;<br/>
<a NAME="E18:24"/>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c4">BX</font> is   <a href="cantor_1.html#NM1" title="CANTOR_1:NM.1">Basis</a> of <font color="Maroon" title="c3">ST</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#T23" title="YELLOW_9:th.23">YELLOW_9:23</a></span>;<br/>
<a NAME="E19:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A33</font></span>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">ST</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c4">BX</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#T22" title="YELLOW_9:th.22">YELLOW_9:22</a></span>;<br/>
<a NAME="E20:24"/>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</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="c5">S9</font>,<font color="Maroon" title="c7">T9</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> 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="c5">S9</font>,<font color="Maroon" title="c7">T9</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="cantor_1.html#T5" title="CANTOR_1:th.5">CANTOR_1:5</a></span>;<br/>
<a NAME="E21:24"/><span class="kw">then </span>
 <a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c4">BX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c5">S9</font>,<font color="Maroon" title="c7">T9</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="txt" href="#E10:24"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="cantor_1.html#T16" title="CANTOR_1:th.16">CANTOR_1:16</a></span>;<br/>
<a NAME="E22:24"/><span class="kw">then </span>
 <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" title="CANTOR_1:func.2">FinMeetCl</a> <font color="Maroon" title="c4">BX</font></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 <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c5">S9</font>,<font color="Maroon" title="c7">T9</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="txt" href="#E9:24"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="ref" href="cantor_1.html#T9" title="CANTOR_1:th.9">CANTOR_1:9</a></span>;<br/>
<a NAME="E23:24"/><span class="kw">then </span>
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">ST</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">S9</font>,<font color="Maroon" title="c7">T9</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="txt" href="#E19:24"><span class="lab"><font color="Green" title="E26">A33</font></span></a>, <a class="txt" href="#E17:24"><span class="lab"><font color="Green" title="E25">A32</font></span></a>, <a class="txt" href="#E15:24"><span class="lab"><font color="Green" title="E24">A31</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:24"/>
 <a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">S9</font>,<font color="Maroon" title="c7">T9</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="waybel19.html#D2" target="_self" title="WAYBEL19:def.2">Def2</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>
