<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">T</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel14.html#D1" target="_self" title="WAYBEL14:def.1">WAYBEL14:def 1</a><br/>

<b>assume </b><a NAME="E2:36"/><i><font color="Green" title="E27">A2</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</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="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font></span>)</span>
 ;<br/>

<a NAME="E3:36"/><i><font color="Green" title="E28">A3</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2" title="WAYBEL11:func.2">Scott-Convergence</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T32" title="WAYBEL11:th.32">WAYBEL11:32</a></i>;<br/>
<a NAME="E4:36"/><i><font color="Green" title="E29">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="yellow_6.html#D27" title="YELLOW_6:def.27">YELLOW_6:def 27</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">Tsup</font> =  <a href="waybel_2.html#K5" title="WAYBEL_2:func.5">sup_op</a> <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E5:36"/><i><font color="Green" title="E30">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</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">T</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>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E6:36"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></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 <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E4:36"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">Tsup</font> =  <a href="waybel_2.html#K5" title="WAYBEL_2:func.5">sup_op</a> <font color="Maroon" title="c1">L</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<font color="Maroon" title="c2">T</font> <b>by </b><i><a class="txt" href="waybel14.html#E4:36"><i><font color="Green" title="E29">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">Tsup</font>
;<br/>

<b>thus </b><a NAME="E8:36"/>
<font color="Maroon" title="c4">Tsup</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="waybel_2.html#K5" title="WAYBEL_2:func.5">sup_op</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<a NAME="E9:36"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> holds  <font color="Maroon" title="c4">Tsup</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">ab</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>;<br/>

<b>consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font><b> such that </b><br/><a NAME="E2:36_1"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c5">ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="waybel14.html#E5:36"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">Tsup</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">ab</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tmap_1.html#D2" title="TMAP_1:def.2">TMAP_1:def 2</a><br/>

<b>reconsider </b><font color="Maroon" title="c9">Tsab</font> = <font color="Maroon" title="c4">Tsup</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">ab</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font> ;<br/>
<a NAME="E4:36_1"/><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c9">Tsab</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c8">G</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E5:36_1"/><i><font color="Green" title="E33">A8</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c8">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T51" title="TOPS_1:th.51">TOPS_1:51</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">Tsab'</font> = <font color="Maroon" title="c9">Tsab</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="yellow_6.html#D27" title="YELLOW_6:def.27">YELLOW_6:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">basab</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Olive" title="b1">q</font></span>)</span> where <font color="Olive" title="b1">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">q</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c10">Tsab'</font> </span>}</span>  as    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c10">Tsab'</font> <b>by </b><i><a class="txt" href="waybel14.html#E1:36"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="waybel11.html#T44" title="WAYBEL11:th.44">WAYBEL11:44</a></i>;<br/>
<a NAME="E8:36_1"/>
<font color="Maroon" title="c11">basab</font> is    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c9">Tsab</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="waybel14.html#E3:36"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="waybel14.html#T21" target="_self" title="WAYBEL14:th.21">Th21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font><b> such that </b><br/><a NAME="E10:36_1"/><i><font color="Green" title="E34">A9</font></i>: 
( <font color="Maroon" title="c12">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">basab</font> &amp; <font color="Maroon" title="c12">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c8">G</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E4:36_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="waybel14.html#E5:36_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E12:36_1"/><i><font color="Green" title="E35">A10</font></i>: 
( <font color="Maroon" title="c12">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c13">u</font> &amp; <font color="Maroon" title="c13">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c10">Tsab'</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E10:36_1"><i><font color="Green" title="E34">A9</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">a'</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c15">b'</font> = <font color="Maroon" title="c7">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="yellow_6.html#D27" title="YELLOW_6:def.27">YELLOW_6:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">Lc</font> = <font color="Maroon" title="c1">L</font> as  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a> <b>by </b><i><a class="txt" href="waybel14.html#E1:36"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E15:36_1"/>
<font color="Maroon" title="c16">Lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">L</font>
 
;<br/>
<a NAME="E16:36_1"/><b>then </b><i><font color="Green" title="E36">A11</font></i>: 
( <font color="Maroon" title="c14">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c14">a'</font></span>)</span> &amp; <font color="Maroon" title="c15">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c15">b'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5" title="WAYBEL_3:def.5">WAYBEL_3:def 5</a></i>;<br/>
<i><font color="Green" title="E37">A12</font></i>: <font color="Maroon" title="c10">Tsab'</font> = 
<font color="Maroon" title="c4">Tsup</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>
<b>by </b><i><a class="txt" href="waybel14.html#E2:36_1"><i><font color="Green" title="E31">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c14">a'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c15">b'</font>
<b>by </b><i><a class="ref" href="waybel_2.html#D5" title="WAYBEL_2:def.5">WAYBEL_2:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c14">a'</font></span>)</span> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c15">b'</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel14.html#E16:36_1"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="waybel_2.html#T4" title="WAYBEL_2:th.4">WAYBEL_2:4</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c17">D1</font> =  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c14">a'</font>;<br/>
<b>set </b><font color="Maroon" title="c18">D2</font> =  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c15">b'</font>;<br/>
<b>set </b><font color="Maroon" title="c19">D</font> = <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c14">a'</font></span>)</span> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c15">b'</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c20">xy</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E19:36_1"/><i><font color="Green" title="E38">A13</font></i>: 
( <font color="Maroon" title="c20">xy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c14">a'</font></span>)</span> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c15">b'</font></span>)</span> &amp; <font color="Maroon" title="c13">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c20">xy</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E1:36"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="waybel14.html#E12:36_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="waybel14.html#E17:36_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="waybel_4.html#T54" title="WAYBEL_4:th.54">WAYBEL_4:54</a></i>;<br/>
<a NAME="E20:36_1"/>
<span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c14">a'</font></span>)</span> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c15">b'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c14">a'</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c15">b'</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#D3" title="YELLOW_4:def.3">YELLOW_4:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">x</font>, <font color="Maroon" title="c22">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E22:36_1"/><i><font color="Green" title="E39">A14</font></i>: 
( <font color="Maroon" title="c20">xy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c22">y</font> &amp; <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c14">a'</font> &amp; <font color="Maroon" title="c22">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c15">b'</font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E19:36_1"><i><font color="Green" title="E38">A13</font></i></a></i>;<br/>
<a NAME="E23:36_1"/><i><font color="Green" title="E40">A15</font></i>: 
( <font color="Maroon" title="c21">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c14">a'</font> &amp; <font color="Maroon" title="c22">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c15">b'</font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E22:36_1"><i><font color="Green" title="E39">A14</font></i></a>, <a class="ref" href="waybel_3.html#T7" title="WAYBEL_3:th.7">WAYBEL_3:7</a></i>;<br/>
<a NAME="E24:36_1"/><i><font color="Green" title="E41">A16</font></i>: 
(  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c21">x</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c22">y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:36"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="waybel11.html#T36" title="WAYBEL11:th.36">WAYBEL11:36</a></i>;<br/>
<a NAME="E25:36_1"/><i><font color="Green" title="E42">A17</font></i>: 
( <font color="Maroon" title="c14">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c21">x</font> &amp; <font color="Maroon" title="c15">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c22">y</font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E23:36_1"><i><font color="Green" title="E40">A15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">wx</font> =  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c21">x</font>, <font color="Maroon" title="c24">wy</font> =  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c22">y</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font> <b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="yellow_6.html#D27" title="YELLOW_6:def.27">YELLOW_6:def 27</a></i>;<br/>
<a NAME="E27:36_1"/>
(  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">L</font> &amp;  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c22">y</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">L</font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E24:36_1"><i><font color="Green" title="E41">A16</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E28:36_1"/><b>then </b><i><font color="Green" title="E43">A18</font></i>: 
( <font color="Maroon" title="c23">wx</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c24">wy</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:36"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="waybel14.html#E3:36"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">H</font> = <span class="p1"><a href="yellow_3.html#K12" title="YELLOW_3:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c21">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c22">y</font></span>)</span></span><a href="yellow_3.html#K12" title="YELLOW_3:func.12">:]</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#K5" title="BORSUK_1:func.5">[:</a><span class="default"><font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <b>by </b><i><a class="txt" href="waybel14.html#E4:36"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="waybel14.html#E5:36"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E30:36_1"/>
<font color="Maroon" title="c25">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E28:36_1"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="borsuk_1.html#T46" title="BORSUK_1:th.46">BORSUK_1:46</a></i>;<br/>
<a NAME="E31:36_1"/><b>then </b><i><font color="Green" title="E44">A19</font></i>: 
<font color="Maroon" title="c25">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c25">H</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/>
<a NAME="E32:36_1"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c14">a'</font>,<font color="Maroon" title="c15">b'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">H</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E25:36_1"><i><font color="Green" title="E42">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">H</font> = <font color="Maroon" title="c25">H</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c5">ab</font> <b>by </b><i><a class="txt" href="waybel14.html#E2:36_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="waybel14.html#E31:36_1"><i><font color="Green" title="E44">A19</font></i></a>, <a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c26">H</font>
;<br/>

<a NAME="E34:36_1"/><i><font color="Green" title="E45">A20</font></i>: 
<font color="Maroon" title="c4">Tsup</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c26">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c21">x</font></span>)</span> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c22">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T35" title="WAYBEL_2:th.35">WAYBEL_2:35</a></i>;<br/>
<a NAME="E35:36_1"/><i><font color="Green" title="E46">A21</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c21">x</font></span>)</span> <a href="yellow_4.html#K2" title="YELLOW_4:func.2">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c22">y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c22">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel14.html#T12" target="_self" title="WAYBEL14:th.12">Th12</a></i>;<br/>
<a NAME="E36:36_1"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c22">y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_3.html#K2" title="WAYBEL_3:func.2">wayabove</a> <font color="Maroon" title="c13">u</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E19:36_1"><i><font color="Green" title="E38">A13</font></i></a>, <a class="txt" href="waybel14.html#E22:36_1"><i><font color="Green" title="E39">A14</font></i></a>, <a class="ref" href="waybel14.html#T7" target="_self" title="WAYBEL14:th.7">Th7</a></i>;<br/>
<a NAME="E37:36_1"/><b>then </b><i><font color="Green" title="E47">A22</font></i>: 
<font color="Maroon" title="c4">Tsup</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c26">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">V</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E12:36_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="waybel14.html#E34:36_1"><i><font color="Green" title="E45">A20</font></i></a>, <a class="txt" href="waybel14.html#E35:36_1"><i><font color="Green" title="E46">A21</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E38:36_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c8">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<a NAME="E39:36_1"/><b>then </b>
<font color="Maroon" title="c12">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G</font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E10:36_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E40:36_1"/>
<font color="Maroon" title="c4">Tsup</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c26">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G</font>
 <b>by </b><i><a class="txt" href="waybel14.html#E37:36_1"><i><font color="Green" title="E47">A22</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:36"/>
<font color="Maroon" title="c4">Tsup</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" title="TMAP_1:th.49">TMAP_1:49</a></i>;<br/>


</div>
