<?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>consider </b><font color="Maroon" title="c2">S</font> being  <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="E2:57"/><i><font color="Green" title="E33">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/>
<a NAME="E3:57"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">S</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="c1">N</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T52" title="YELLOW_9:th.52">YELLOW_9:52</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon" title="c1">N</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 ;<br/><a NAME="E2:57_1"/><b>then </b><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="yellow12.html#T15" title="YELLOW12:th.15">YELLOW12:15</a></i>;<br/><a NAME="E3:57_1"/>
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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_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="c2">S</font>;<br/>

<b>consider </b><font color="Maroon" title="c4">J</font> being    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c3">x</font><b> such that </b><br/><a NAME="E2:57_1_1"/><i><font color="Green" title="E36">A4</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="c2">S</font>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">J</font> holds <br/>( <font color="Olive" title="b1">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">W</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> )
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="waybel14.html#T35" title="WAYBEL14:th.35">WAYBEL14:35</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">J</font>
;<br/>

<b>let </b><font color="Maroon" title="c5">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="E3:57_1_1"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c5">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">J</font>
 ;<br/>

<a NAME="E4:57_1_1"/><b>then </b>
<font color="Maroon" title="c5">W</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#E2:57_1_1"><i><font color="Green" title="E36">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:57_1_1"/>
<font color="Maroon" title="c5">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#E2:57_1_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="waybel30.html#E3:57_1_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:57_1"/>
<font color="Maroon" title="c1">N</font> is <a href="waybel30.html#V3" title="WAYBEL30:attr.3">with_open_semilattices</a>
 <b>by </b><i><a class="ref" href="waybel30.html#T32" target="_self" title="WAYBEL30:th.32">Th32</a></i>;<br/><a NAME="E5:57_1"/>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c2">S</font></span>)</span> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="waybel14.html#T36" title="WAYBEL14:th.36">WAYBEL14:36</a></i>;<br/><b>hence </b><a NAME="E6:57_1"/>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="yellow_9.html#T52" title="YELLOW_9:th.52">YELLOW_9:52</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:57"/><i><font color="Green" title="E35">A6</font></i>: 
( <font color="Maroon" title="c1">N</font> is <a href="waybel30.html#V3" title="WAYBEL30:attr.3">with_open_semilattices</a> &amp;  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> )
 ;<br/>

<a NAME="E6:57"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">Y</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">Y</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">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">Y</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="txt" href="waybel30.html#E2:57"><i><font color="Green" title="E33">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="E3:57_2"/><i><font color="Green" title="E36">A7</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#E5:57"><i><font color="Green" title="E35">A6</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">Y</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="E5:57_2"/><i><font color="Green" title="E37">A8</font></i>: 
<font color="Maroon" title="c7">Y</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">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="E7:57_2"/><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon" title="c7">Y</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">A</font>
 <b>and </b><br/><a NAME="E8:57_2"/><i><font color="Green" title="E39">A10</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font>
 ;<br/>
<a NAME="E9:57_2"/>
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c8">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#E3:57_2"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="waybel30.html#E8:57_2"><i><font color="Green" title="E39">A10</font></i></a></i>;<br/>
<a NAME="E10:57_2"/><b>then </b>
<font color="Maroon" title="c8">A</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="E11:57_2"/><b>then </b>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c8">A</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="E12:57_2"/>
( <font color="Maroon" title="c7">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c7">Y</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> )
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="waybel30.html#E5:57_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="waybel30.html#E7:57_2"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="waybel_0.html#T4" title="WAYBEL_0:th.4">WAYBEL_0:4</a>, <a class="ref" href="yellow_8.html#T21" title="YELLOW_8:th.21">YELLOW_8:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:57"/><b>then </b>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> holds  <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Olive" title="b2">X</font></span>)</span> where <font color="Olive" title="b2">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b2">X</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> ) </span>}</span> ,<font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="waybel30.html#E3:57"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="waybel30.html#E5:57"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="waybel14.html#T37" title="WAYBEL14:th.37">WAYBEL14:37</a></i>;<br/>
<a NAME="E8:57"/><b>then </b>
<font color="Maroon" title="c2">S</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel14.html#T38" title="WAYBEL14:th.38">WAYBEL14:38</a></i>;<br/>
<b>hence </b><a NAME="E9:57"/>
<font color="Maroon" title="c1">N</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel30.html#E2:57"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="yellow12.html#T15" title="YELLOW12:th.15">YELLOW12:15</a></i>;<br/>


</div>
