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

<b>let </b><font color="Maroon" title="c1">R1</font>, <font color="Maroon" title="c2">R2</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:95"/><i><font color="Green" title="E64">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">R1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R1</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">R2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">R2</font> #)
 ;<br/>

<b>assume </b><a NAME="E2:95"/>
<font color="Maroon" title="c1">R1</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 ;<br/>

<a NAME="E3:95"/><b>then </b><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon" title="c1">R1</font> is <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D6" title="WAYBEL_3:def.6">WAYBEL_3:def 6</a></i>;<br/>
<b>thus </b><a NAME="E4:95"/><i><font color="Green" title="E66">A3</font></i>: 
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">R2</font> holds <br/> ( not  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D6" title="WAYBEL_3:def.6">WAYBEL_3:def 6</a><br/>

<b>thus </b><a NAME="E5:95"/>
<font color="Maroon" title="c2">R2</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a>
 ;<br/>

<a NAME="E6:95"/><i><font color="Green" title="E67">A4</font></i>: 
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="c1">R1</font> holds <br/> ( not  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
;<br/>
<a NAME="E7:95"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font>  st not <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/> ex <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font> st <br/>( <font color="Olive" title="b3">u</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x</font> &amp; not <font color="Olive" title="b3">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">y2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">y2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font> <b>by </b><i><a class="txt" href="waybel11.html#E1:95"><i><font color="Green" title="E64">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:95_1"/>
not <font color="Maroon" title="c3">x2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">y2</font>
 ;<br/>

<a NAME="E3:95_1"/><b>then </b>
not <font color="Maroon" title="c5">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y1</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:95"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">u1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R1</font><b> such that </b><br/><a NAME="E5:95_1"/><i><font color="Green" title="E68">A5</font></i>: 
<font color="Maroon" title="c7">u1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c5">x1</font>
 <b>and </b><br/><a NAME="E6:95_1"/><i><font color="Green" title="E69">A6</font></i>: 
not <font color="Maroon" title="c7">u1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y1</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E3:95"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="waybel11.html#E6:95"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="waybel_3.html#T24" title="WAYBEL_3:th.24">WAYBEL_3:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">u2</font> = <font color="Maroon" title="c7">u1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">R2</font> <b>by </b><i><a class="txt" href="waybel11.html#E1:95"><i><font color="Green" title="E64">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">u2</font>
;<br/>

<b>thus </b><a NAME="E8:95_1"/>
<font color="Maroon" title="c8">u2</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">x2</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:95"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="waybel11.html#E5:95_1"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="waybel11.html#E64"><i><font color="Green" title="E55">Lm15</font></i></a></i>;<br/>

<b>thus </b><a NAME="E9:95_1"/>
not <font color="Maroon" title="c8">u2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">y2</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E1:95"><i><font color="Green" title="E64">A1</font></i></a>, <a class="txt" href="waybel11.html#E6:95_1"><i><font color="Green" title="E69">A6</font></i></a>, <a class="txt" href="waybel11.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:95"/>
<font color="Maroon" title="c2">R2</font> is <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a>
 <b>by </b><i><a class="txt" href="waybel11.html#E4:95"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="waybel_3.html#T24" title="WAYBEL_3:th.24">WAYBEL_3:24</a></i>;<br/>


</div>
