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

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:104_1"/><i><font color="Green" title="E62">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 ;<br/><b>hence </b><a NAME="E2:104_1"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>
 <b>by </b><i><a class="ref" href="waybel_4.html#T38" target="_self" title="WAYBEL_4:th.38">Th38</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c2">R</font> = <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  as  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel_4.html#E1:104_1"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="waybel_4.html#E72"><i><font color="Green" title="E56">Lm12</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c3">R</font> = <font color="Maroon" title="c2">R</font>;<br/><b>thus </b><a NAME="E4:104_1"/>
for <font color="Olive" title="b1">R'</font> being <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c3">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">R'</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E1:104_1"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="waybel_4.html#T45" target="_self" title="WAYBEL_4:th.45">Th45</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:104"/><i><font color="Green" title="E62">A2</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>
 ;<br/>

<b>given </b><font color="Maroon" title="c2">R</font> being  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font><b> such that </b><a NAME="E4:104"/><i><font color="Green" title="E63">A3</font></i>: 
for <font color="Olive" title="b1">R'</font> being <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">R'</font>
 ;<br/>

<a NAME="E5:104"/>
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">L</font> holds  <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_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="c1">L</font>;<br/>

<b>set </b><font color="Maroon" title="c4">K</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font></span>)</span> where <font color="Olive" title="b1">AR</font> is  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">AR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K12" title="WAYBEL_4:func.12">App</a> <font color="Maroon" title="c1">L</font> </span>}</span> ;<br/>
<a NAME="E1:104_2"/><i><font color="Green" title="E64">A4</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font></span>)</span> where <font color="Olive" title="b1">AR</font> is  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">AR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K12" title="WAYBEL_4:func.12">App</a> <font color="Maroon" title="c1">L</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E2:104"><i><font color="Green" title="E62">A2</font></i></a>, <a class="ref" href="waybel_4.html#T44" target="_self" title="WAYBEL_4:th.44">Th44</a></i>;<br/>
<a NAME="E2:104_2"/>
<font color="Maroon" title="c2">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K12" title="WAYBEL_4:func.12">App</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D20" target="_self" title="WAYBEL_4:def.20">Def20</a></i>;<br/>
<a NAME="E3:104_2"/><b>then </b><i><font color="Green" title="E65">A5</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font></span>)</span> where <font color="Olive" title="b1">AR</font> is  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">AR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K12" title="WAYBEL_4:func.12">App</a> <font color="Maroon" title="c1">L</font> </span>}</span> 
 
;<br/>
<a NAME="E4:104_2"/><b>then </b><i><font color="Green" title="E66">A6</font></i>: 
 <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">R</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E1:104_2"><i><font color="Green" title="E64">A4</font></i></a>, <a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>
<a NAME="E5:104_2"/><i><font color="Green" title="E67">A7</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D18" target="_self" title="WAYBEL_4:def.18">Def18</a></i>;<br/>
<a NAME="E6:104_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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font></span>)</span> where <font color="Olive" title="b2">AR</font> is  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b2">AR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K12" title="WAYBEL_4:func.12">App</a> <font color="Maroon" title="c1">L</font> </span>}</span>  holds <br/><font color="Maroon" title="c2">R</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:104_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font></span>)</span> where <font color="Olive" title="b1">AR</font> is  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">AR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K12" title="WAYBEL_4:func.12">App</a> <font color="Maroon" title="c1">L</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">AA</font> being  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:104_2_1"/><i><font color="Green" title="E68">A8</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">AA</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c6">AA</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K12" title="WAYBEL_4:func.12">App</a> <font color="Maroon" title="c1">L</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">AA</font> = <font color="Maroon" title="c6">AA</font> as  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel_4.html#E3:104_2_1"><i><font color="Green" title="E68">A8</font></i></a>, <a class="ref" href="waybel_4.html#D20" target="_self" title="WAYBEL_4:def.20">Def20</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">b</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="E5:104_2_1"/><i><font color="Green" title="E69">A9</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font>
 ;<br/>

<a NAME="E6:104_2_1"/>
<font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c7">AA</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E4:104"><i><font color="Green" title="E63">A3</font></i></a></i>;<br/>
<a NAME="E7:104_2_1"/><b>then </b>
<font color="Maroon" title="c2">R</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">AA</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T29" target="_self" title="WAYBEL_4:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E8:104_2_1"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E3:104_2_1"><i><font color="Green" title="E68">A8</font></i></a>, <a class="txt" href="waybel_4.html#E5:104_2_1"><i><font color="Green" title="E69">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:104_2"/><b>then </b>
<font color="Maroon" title="c2">R</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font></span>)</span> where <font color="Olive" title="b1">AR</font> is  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">AR</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K12" title="WAYBEL_4:func.12">App</a> <font color="Maroon" title="c1">L</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel_4.html#E3:104_2"><i><font color="Green" title="E65">A5</font></i></a>, <a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></i>;<br/>
<b>hence </b><a NAME="E8:104_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c3">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_4.html#E1:104_2"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="waybel_4.html#E4:104_2"><i><font color="Green" title="E66">A6</font></i></a>, <a class="txt" href="waybel_4.html#E5:104_2"><i><font color="Green" title="E67">A7</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:104"/><b>then </b><i><font color="Green" title="E64">A10</font></i>: 
<font color="Maroon" title="c1">L</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#D5" title="WAYBEL_3:def.5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E7:104"/>
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">L</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/>
<b>hence </b><a NAME="E8:104"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel_4.html#E6:104"><i><font color="Green" title="E64">A10</font></i></a>, <a class="ref" href="waybel_3.html#D6" title="WAYBEL_3:def.6">WAYBEL_3:def 6</a></i>;<br/>


</div>
