<?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/>



<a NAME="E1:54"/><i><font color="Green" title="E30">A1</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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_1"/><i><font color="Green" title="E31">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font>  ex <font color="Olive" title="b2">J</font> being   <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Olive" title="b1">x</font> st <br/>for <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b3">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">J</font> holds <br/><font color="Olive" title="b3">W</font> is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c2">S</font>
 ;<br/><b>thus </b><a NAME="E2:54_1"/>
<font color="Maroon" title="c1">N</font> is <a href="waybel30.html#V3" title="WAYBEL30:attr.3">with_open_semilattices</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>proof </b></a><div class="add">

<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="c1">N</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel30.html#D4" target="_self" title="WAYBEL30:def.4">WAYBEL30:def 4</a><br/>

<div><i><font color="Green" title="E32">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:54_1_1_1"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c4">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:54_1_1_1"/><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c5">BL</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">O</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b2">F</font></span>)</span></span>)</span> where <font color="Olive" title="b1">O</font>, <font color="Olive" title="b2">F</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">O</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="c1">N</font> &amp; <font color="Olive" title="b2">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) </span>}</span>  as    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="ref" href="waybel19.html#T32" title="WAYBEL19:th.32">WAYBEL19:32</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c3">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel30.html#E1:54"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">By</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c6">y</font><b> such that </b><br/><a NAME="E6:54_1_1_1"/><i><font color="Green" title="E35">A6</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">By</font> holds <br/><font color="Olive" title="b1">A</font> is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E1:54_1"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/><a NAME="E7:54_1_1_1"/>
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</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">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">BL</font> &amp; <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">W</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel30.html#E1:54_1_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="yellow_8.html#T18" title="YELLOW_8:th.18">YELLOW_8:18</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">K</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:54_1_1_1"/><i><font color="Green" title="E36">A7</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">K</font> &amp; <font color="Maroon" title="c8">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</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">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">BL</font> &amp; <font color="Olive" title="b1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">W</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="waybel30.html#E2:54_1_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">G</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="E11:54_1_1_1"/><i><font color="Green" title="E37">A8</font></i>: 
<font color="Maroon" title="c8">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">G</font>
 <b>and </b><br/><a NAME="E12:54_1_1_1"/><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon" title="c9">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">BL</font>
 <b>and </b><br/><a NAME="E13:54_1_1_1"/><i><font color="Green" title="E39">A10</font></i>: 
<font color="Maroon" title="c9">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E9:54_1_1_1"><i><font color="Green" title="E36">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c10">V</font>, <font color="Maroon" title="c11">F</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="E15:54_1_1_1"/><i><font color="Green" title="E40">A11</font></i>: 
<font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">V</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c11">F</font></span>)</span>
 <b>and </b><br/><a NAME="E16:54_1_1_1"/><i><font color="Green" title="E41">A12</font></i>: 
<font color="Maroon" title="c10">V</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="c1">N</font>
 <b>and </b><br/><a NAME="E17:54_1_1_1"/><i><font color="Green" title="E42">A13</font></i>: 
<font color="Maroon" title="c11">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E12:54_1_1_1"><i><font color="Green" title="E38">A9</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">V</font> = <font color="Maroon" title="c10">V</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#E1:54"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/><a NAME="E19:54_1_1_1"/><i><font color="Green" title="E43">A14</font></i>: 
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="txt" href="waybel30.html#E1:54"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="yellow_9.html#T52" title="YELLOW_9:th.52">YELLOW_9:52</a></i>;<br/><a NAME="E20:54_1_1_1"/><b>then </b><i><font color="Green" title="E44">A15</font></i>: 
<font color="Maroon" title="c12">V</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#E16:54_1_1_1"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="waybel14.html#T24" title="WAYBEL14:th.24">WAYBEL14:24</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">F</font> = <font color="Maroon" title="c11">F</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <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#E17:54_1_1_1"><i><font color="Green" title="E42">A13</font></i></a></i>;<br/><a NAME="E22:54_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">V</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E9:54_1_1_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="waybel30.html#E11:54_1_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="waybel30.html#E15:54_1_1_1"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">U1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E24:54_1_1_1"/><i><font color="Green" title="E45">A16</font></i>: 
<font color="Maroon" title="c14">U1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">By</font>
 <b>and </b><br/><a NAME="E25:54_1_1_1"/><i><font color="Green" title="E46">A17</font></i>: 
<font color="Maroon" title="c14">U1</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="waybel30.html#E20:54_1_1_1"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="yellow_8.html#T22" title="YELLOW_8:th.22">YELLOW_8:22</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">U2</font> = <font color="Maroon" title="c14">U1</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#E1:54"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/><a NAME="E27:54_1_1_1"/>
<font color="Maroon" title="c14">U1</font> is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E6:54_1_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="waybel30.html#E24:54_1_1_1"><i><font color="Green" title="E45">A16</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">U2</font> = <font color="Maroon" title="c15">U2</font> as   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="txt" href="waybel30.html#E1:54"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="waybel_0.html#T4" title="WAYBEL_0:th.4">WAYBEL_0:4</a>, <a class="ref" href="waybel_0.html#T25" title="WAYBEL_0:th.25">WAYBEL_0:25</a></i>;<br/><a NAME="E29:54_1_1_1"/>
<font color="Maroon" title="c16">U2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c13">F</font></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c17">IT</font> = <font color="Maroon" title="c14">U1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c13">F</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> ;<br/><b>take </b><font color="Maroon" title="c18">U2</font> = <font color="Maroon" title="c16">U2</font>;<br/><b>take </b><font color="Maroon" title="c19">F</font> = <font color="Maroon" title="c13">F</font>;<br/><b>take </b><font color="Maroon" title="c20">IT</font> = <font color="Maroon" title="c17">IT</font>;<br/><b>thus </b><a NAME="E31:54_1_1_1"/>
<font color="Maroon" title="c20">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">U2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c19">F</font></span>)</span>
 ;<br/><a NAME="E32:54_1_1_1"/><i><font color="Green" title="E47">A18</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">U1</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E24:54_1_1_1"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>;<br/><a NAME="E33:54_1_1_1"/>
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c19">F</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E9:54_1_1_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="waybel30.html#E11:54_1_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="waybel30.html#E15:54_1_1_1"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E34:54_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">IT</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E32:54_1_1_1"><i><font color="Green" title="E47">A18</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E35:54_1_1_1"/>
<font color="Maroon" title="c14">U1</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#E24:54_1_1_1"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>;<br/><a NAME="E36:54_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">U1</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="waybel14.html#T24" title="WAYBEL14:th.24">WAYBEL14:24</a></i>;<br/><a NAME="E37:54_1_1_1"/><b>then </b><i><font color="Green" title="E48">A19</font></i>: 
<font color="Maroon" title="c20">IT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">BL</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E19:54_1_1_1"><i><font color="Green" title="E43">A14</font></i></a></i>;<br/><a NAME="E38:54_1_1_1"/>
<font color="Maroon" title="c5">BL</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="c1">N</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#D2" title="CANTOR_1:def.2">CANTOR_1:def 2</a></i>;<br/><b>hence </b><a NAME="E39:54_1_1_1"/>
<font color="Maroon" title="c20">IT</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#E37:54_1_1_1"><i><font color="Green" title="E48">A19</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="E40:54_1_1_1"/>
<font color="Maroon" title="c20">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">G</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E15:54_1_1_1"><i><font color="Green" title="E40">A11</font></i></a>, <a class="txt" href="waybel30.html#E25:54_1_1_1"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></i>;<br/><b>hence </b><a NAME="E41:54_1_1_1"/>
<font color="Maroon" title="c20">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E13:54_1_1_1"><i><font color="Green" title="E39">A10</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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">U1</font> being  <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">N</font> ex <font color="Olive" title="b2">F</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> ex <font color="Olive" title="b3">W1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">W1</font> &amp; <font color="Olive" title="b1">U1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; <font color="Olive" title="b3">W1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> );<br/>
<b>consider </b><font color="Maroon" title="c4">SF</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E3:54_1_1"/><i><font color="Green" title="E33">A20</font></i>: 
for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> holds <br/> ( <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">SF</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">W</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">SF</font> = <font color="Maroon" title="c4">SF</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">N</font> ;<br/>
<a NAME="E5:54_1_1"/>
<font color="Maroon" title="c5">SF</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="54_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:54_1_1_2"/>
<font color="Maroon" title="c5">SF</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="c1">N</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="54_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</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="E1:54_1_1_2_1"/><i><font color="Green" title="E34">A21</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SF</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">W</font> = <font color="Maroon" title="c6">a</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> ;<br/>
<b>consider </b><font color="Maroon" title="c8">U1</font> being   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">N</font>, <font color="Maroon" title="c9">F</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>, <font color="Maroon" title="c10">W1</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="E4:54_1_1_2_1"/><i><font color="Green" title="E35">A22</font></i>: 
<font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font>
 <b>and </b><br/><a NAME="E5:54_1_1_2_1"/>
( <font color="Maroon" title="c8">U1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> )
 <b>and </b><br/><a NAME="E6:54_1_1_2_1"/><i><font color="Green" title="E36">A23</font></i>: 
<font color="Maroon" title="c10">W1</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#E3:54_1_1"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="waybel30.html#E1:54_1_1_2_1"><i><font color="Green" title="E34">A21</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:54_1_1_2_1"/>
<font color="Maroon" title="c6">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="c1">N</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E4:54_1_1_2_1"><i><font color="Green" title="E35">A22</font></i></a>, <a class="txt" href="waybel30.html#E6:54_1_1_2_1"><i><font color="Green" title="E36">A23</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


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

<a NAME="E2:54_1_1_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SF</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">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1_1_2_2"/><i><font color="Green" title="E34">A24</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SF</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">W</font> = <font color="Maroon" title="c6">a</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> ;<br/>
<b>consider </b><font color="Maroon" title="c8">U1</font> being   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">N</font>, <font color="Maroon" title="c9">F</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>, <font color="Maroon" title="c10">W1</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="E4:54_1_1_2_2"/><i><font color="Green" title="E35">A25</font></i>: 
( <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c8">U1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c10">W1</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#E3:54_1_1"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="waybel30.html#E1:54_1_1_2_2"><i><font color="Green" title="E34">A24</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:54_1_1_2_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E4:54_1_1_2_2"><i><font color="Green" title="E35">A25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:54_1_1_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="c5">SF</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="c6">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>;<br/>

<b>assume </b><a NAME="E4:54_1_1_2"/>
( <font color="Maroon" title="c6">W</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="c6">W</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">U2</font> being   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">N</font>, <font color="Maroon" title="c8">F</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>, <font color="Maroon" title="c9">IT</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="E6:54_1_1_2"/><i><font color="Green" title="E34">A26</font></i>: 
( <font color="Maroon" title="c9">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">U2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c8">F</font></span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">IT</font> &amp; <font color="Maroon" title="c9">IT</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c9">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W</font> )
 <b>by </b><i><a class="txt" href="waybel30.html#E1:54_1_1"><i><font color="Green" title="E32">A3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">IT</font>
;<br/>

<b>thus </b><a NAME="E7:54_1_1_2"/>
( <font color="Maroon" title="c9">IT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">SF</font> &amp; <font color="Maroon" title="c9">IT</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">W</font> )
 <b>by </b><i><a class="txt" href="waybel30.html#E3:54_1_1"><i><font color="Green" title="E33">A20</font></i></a>, <a class="txt" href="waybel30.html#E6:54_1_1_2"><i><font color="Green" title="E34">A26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">SF</font> = <font color="Maroon" title="c5">SF</font> as    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">x</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">SF</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>;<br/>

<b>assume </b><a NAME="E7:54_1_1"/>
<font color="Maroon" title="c7">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">SF</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">U1</font> being   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">N</font>, <font color="Maroon" title="c9">F</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>, <font color="Maroon" title="c10">W1</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="E9:54_1_1"/><i><font color="Green" title="E34">A27</font></i>: 
( <font color="Maroon" title="c10">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c8">U1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c10">W1</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#E3:54_1_1"><i><font color="Green" title="E33">A20</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">SW</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">W</font>;<br/>
<b>thus </b><a NAME="E10:54_1_1"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">W</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D16" title="YELLOW_0:def.16">YELLOW_0:def 16</a><br/>

<b>assume </b><a NAME="E1:54_1_1_3"/><i><font color="Green" title="E35">A28</font></i>: 
( <font color="Maroon" title="c12">a</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">W</font></span>)</span> &amp; <font color="Maroon" title="c13">b</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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">W</font></span>)</span> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">N</font> )
 ;<br/>

<a NAME="E2:54_1_1_3"/><i><font color="Green" title="E36">A29</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">W</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E3:54_1_1_3"/><b>then </b><i><font color="Green" title="E37">A30</font></i>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">U1</font> &amp; <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">U1</font> &amp; not <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">F</font> &amp; not <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">F</font> )
 
<b>by </b><i><a class="txt" href="waybel30.html#E9:54_1_1"><i><font color="Green" title="E34">A27</font></i></a>, <a class="txt" href="waybel30.html#E1:54_1_1_3"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E5:54_1_1_3"/><i><font color="Green" title="E38">A31</font></i>: 
( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">U1</font> &amp; <font color="Maroon" title="c14">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c14">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">b</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a></i>;<br/>
<a NAME="E6:54_1_1_3"/>
<font color="Maroon" title="c14">z</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c14">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">b</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E5:54_1_1_3"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="yellow_3.html#T2" title="YELLOW_3:th.2">YELLOW_3:2</a></i>;<br/>
<a NAME="E7:54_1_1_3"/><b>then </b>
<font color="Maroon" title="c14">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">b</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></i>;<br/>
<a NAME="E8:54_1_1_3"/><b>then </b><i><font color="Green" title="E39">A32</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">U1</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E5:54_1_1_3"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></i>;<br/>
<a NAME="E9:54_1_1_3"/>
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st <font color="Olive" title="b1">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">b</font> holds <br/> not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>;<br/>

<b>assume </b><a NAME="E1:54_1_1_3_1"/><i><font color="Green" title="E40">A33</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">b</font>
 ;<br/>

<a NAME="E2:54_1_1_3_1"/>
<font color="Maroon" title="c12">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">a</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<a NAME="E3:54_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c15">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">a</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E1:54_1_1_3_1"><i><font color="Green" title="E40">A33</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E4:54_1_1_3_1"/>
not <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">F</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E3:54_1_1_3"><i><font color="Green" title="E37">A30</font></i></a>, <a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:54_1_1_3"/><b>then </b>
not <font color="Maroon" title="c12">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c9">F</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></i>;<br/>
<a NAME="E11:54_1_1_3"/><b>then </b>
<font color="Maroon" title="c12">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E9:54_1_1"><i><font color="Green" title="E34">A27</font></i></a>, <a class="txt" href="waybel30.html#E8:54_1_1_3"><i><font color="Green" title="E39">A32</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:54_1_1_3"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <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 <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c7">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:54_1_1_3"><i><font color="Green" title="E36">A29</font></i></a>, <a class="ref" href="yellow_0.html#T40" title="YELLOW_0:th.40">YELLOW_0:40</a></i>;<br/>


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


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

<b>assume </b><a NAME="E3:54"/><i><font color="Green" title="E31">A34</font></i>: 
<font color="Maroon" title="c1">N</font> is <a href="waybel30.html#V3" title="WAYBEL30:attr.3">with_open_semilattices</a>
 ;<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>reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="txt" href="waybel30.html#E1:54"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">J</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c4">y</font><b> such that </b><br/><a NAME="E6:54"/><i><font color="Green" title="E32">A35</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>  st <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> holds <br/> <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">A</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E3:54"><i><font color="Green" title="E31">A34</font></i></a>, <a class="ref" href="waybel30.html#D4" target="_self" title="WAYBEL30:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">J'</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="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">x</font> <b>by </b><i><a class="ref" href="waybel30.html#T16" target="_self" title="WAYBEL30:th.16">Th16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">J'</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">W</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="E8:54"/>
<font color="Maroon" title="c7">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">V</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="E10:54"/><i><font color="Green" title="E33">A36</font></i>: 
<font color="Maroon" title="c7">W</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="c8">V</font>
 <b>and </b><br/><a NAME="E11:54"/><i><font color="Green" title="E34">A37</font></i>: 
<font color="Maroon" title="c8">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 ;<br/>
<a NAME="E12:54"/><i><font color="Green" title="E35">A38</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E11:54"><i><font color="Green" title="E34">A37</font></i></a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>;<br/>
<a NAME="E13:54"/><i><font color="Green" title="E36">A39</font></i>: 
<font color="Maroon" title="c8">V</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="c8">V</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T16" title="WAYBEL_0:th.16">WAYBEL_0:16</a></i>;<br/>
<a NAME="E14:54"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c8">V</font> is <a href="yellow_0.html#V5" title="YELLOW_0:attr.5">meet-inheriting</a>
 
<b>by </b><i><a class="txt" href="waybel30.html#E6:54"><i><font color="Green" title="E32">A35</font></i></a>, <a class="txt" href="waybel30.html#E11:54"><i><font color="Green" title="E34">A37</font></i></a></i>;<br/>
<a NAME="E15:54"/><b>then </b>
<font color="Maroon" title="c8">V</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a>
 
<b>by </b><i><a class="ref" href="yellow12.html#T26" title="YELLOW12:th.26">YELLOW12:26</a></i>;<br/>
<a NAME="E16:54"/><b>then </b>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c8">V</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T35" title="WAYBEL_0:th.35">WAYBEL_0:35</a></i>;<br/>
<b>hence </b><a NAME="E17:54"/>
<font color="Maroon" title="c7">W</font> is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="waybel30.html#E1:54"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="waybel30.html#E10:54"><i><font color="Green" title="E33">A36</font></i></a>, <a class="txt" href="waybel30.html#E12:54"><i><font color="Green" title="E35">A38</font></i></a>, <a class="txt" href="waybel30.html#E13:54"><i><font color="Green" title="E36">A39</font></i></a>, <a class="ref" href="waybel_0.html#T4" title="WAYBEL_0:th.4">WAYBEL_0:4</a>, <a class="ref" href="waybel_0.html#T25" title="WAYBEL_0:th.25">WAYBEL_0:25</a></i>;<br/>


</div>
