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

<b>let </b><font color="Maroon" title="c1">N</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a> <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">N</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font>;<br/><b>let </b><font color="Maroon" title="c5">J</font> be    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c4">y</font>;<br/>









<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 ;<br/>

<a NAME="E2:30"/><i><font color="Green" title="E16">A2</font></i>: 
 <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">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">N</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">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">Z</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c7">K</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> </span>}</span> ;<br/>
<a NAME="E3:30"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> </span>}</span>  is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E3:30_1_1"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">A</font>
 <b>and </b><br/><a NAME="E4:30_1_1"/>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 ;<br/>
<b>thus </b><a NAME="E5:30_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:30"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="waybel30.html#E3:30_1_1"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:30_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> </span>}</span>  is   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">S</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">K</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E5:30"/>
<font color="Maroon" title="c8">K</font> is    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_2"/>
<font color="Maroon" title="c8">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_2_1"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E3:30_2_1"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c10">A</font>
 <b>and </b><br/><a NAME="E4:30_2_1"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">A'</font> = <font color="Maroon" title="c10">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel30.html#E2:30"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<a NAME="E6:30_2_1"/>
<font color="Maroon" title="c10">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel30.html#E4:30_2_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>;<br/>
<a NAME="E7:30_2_1"/><b>then </b>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">A'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="waybel30.html#T15" target="_self" title="WAYBEL30:th.15">Th15</a></i>;<br/>
<a NAME="E8:30_2_1"/><b>then </b>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">A'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:30_2_1"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:30"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="waybel30.html#E3:30_2_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E2:30_2"/>
for <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K</font> holds <br/><font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:30_2_2"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E3:30_2_2"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c10">A</font>
 <b>and </b><br/><a NAME="E4:30_2_2"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 ;<br/>
<a NAME="E5:30_2_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c5">J</font>
 
<b>by </b><i><a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>
<a NAME="E6:30_2_2"/><b>then </b><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E4:30_2_2"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/>
<a NAME="E7:30_2_2"/>
<font color="Maroon" title="c10">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c10">A</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T16" title="WAYBEL_0:th.16">WAYBEL_0:16</a></i>;<br/>
<b>hence </b><a NAME="E8:30_2_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E1:30"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="waybel30.html#E3:30_2_2"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="waybel30.html#E6:30_2_2"><i><font color="Green" title="E19">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c8">K</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/>

<b>let </b><font color="Maroon" title="c9">sA</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><a NAME="E4:30_2"/><i><font color="Green" title="E17">A9</font></i>: 
( <font color="Maroon" title="c9">sA</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">sA</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">lA</font> = <font color="Maroon" title="c9">sA</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="txt" href="waybel30.html#E2:30"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<a NAME="E6:30_2"/>
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <font color="Maroon" title="c1">N</font>
 
<b>by </b><i><a class="ref" href="waybel30.html#T10" target="_self" title="WAYBEL30:th.10">Th10</a></i>;<br/>
<a NAME="E7:30_2"/><b>then </b><i><font color="Green" title="E18">A10</font></i>: 
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <font color="Maroon" title="c1">N</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E2:30"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="yellow_9.html#T52" title="YELLOW_9:th.52">YELLOW_9:52</a></i>;<br/>
<a NAME="E8:30_2"/>
<font color="Maroon" title="c1">N</font> is  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a> <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E2:30"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E9:30_2"/><b>then </b>
<font color="Maroon" title="c10">lA</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel30.html#E4:30_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="waybel19.html#T37" title="WAYBEL19:th.37">WAYBEL19:37</a></i>;<br/>
<a NAME="E10:30_2"/><b>then </b>
<font color="Maroon" title="c10">lA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <font color="Maroon" title="c1">N</font>
 
<b>by </b><i><a class="ref" href="waybel30.html#T12" target="_self" title="WAYBEL30:th.12">Th12</a></i>;<br/>
<a NAME="E11:30_2"/><b>then </b>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c10">lA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="waybel30.html#T14" target="_self" title="WAYBEL30:th.14">Th14</a></i>;<br/>
<a NAME="E12:30_2"/><b>then </b><i><font color="Green" title="E19">A11</font></i>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c10">lA</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel30.html#E7:30_2"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="waybel30.html#T12" target="_self" title="WAYBEL30:th.12">Th12</a></i>;<br/>
<a NAME="E13:30_2"/>
<font color="Maroon" title="c10">lA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c10">lA</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T16" title="WAYBEL_0:th.16">WAYBEL_0:16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">lV1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E15:30_2"/><i><font color="Green" title="E20">A12</font></i>: 
<font color="Maroon" title="c11">lV1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 <b>and </b><br/><a NAME="E16:30_2"/><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c11">lV1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c10">lA</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E1:30"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="waybel30.html#E4:30_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="waybel30.html#E12:30_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">sUV</font> =  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">lV1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel30.html#E2:30"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">sUV</font>
;<br/>

<b>thus </b><a NAME="E18:30_2"/>
<font color="Maroon" title="c12">sUV</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E15:30_2"><i><font color="Green" title="E20">A12</font></i></a></i>;<br/>

<a NAME="E19:30_2"/><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c9">sA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">sA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_2_3"/>
<font color="Maroon" title="c9">sA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">sA</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T16" title="WAYBEL_0:th.16">WAYBEL_0:16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<a NAME="E2:30_2_3"/>
<font color="Maroon" title="c9">sA</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 
<b>by </b><i><a class="txt" href="waybel30.html#E4:30_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:30_2_3"/>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">sA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">sA</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T24" title="WAYBEL_0:th.24">WAYBEL_0:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:30_2"/><i><font color="Green" title="E23">A15</font></i>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">sA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c10">lA</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E2:30"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></i>;<br/>
<a NAME="E21:30_2"/>
<font color="Maroon" title="c11">lV1</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c10">lA</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E16:30_2"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="waybel12.html#T20" title="WAYBEL12:th.20">WAYBEL12:20</a></i>;<br/>
<b>hence </b><a NAME="E22:30_2"/>
<font color="Maroon" title="c12">sUV</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">sA</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E19:30_2"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="waybel30.html#E20:30_2"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="yellow12.html#T28" title="YELLOW12:th.28">YELLOW12:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:30"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> </span>}</span>  is    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">x</font>
 ;<br/>


</div>
