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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:104_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/><b>hence </b><a NAME="E2:104_1"/>
<font color="Maroon">L</font> is <a href="waybel_2.html#V2">meet-continuous</a>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">L</font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  as  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel_4.html#T4" target="_self">Th4</a>, <a class="ref" href="waybel_4.html#D9" target="_self">Def9</a></i>;<br/><b>take </b><font color="Maroon">R</font> = <font color="Maroon">R</font>;<br/><b>thus </b><a NAME="E4:104_1"/>
for <font color="Olive">R'</font> being <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> holds  <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Olive">R'</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#T4" target="_self">Th4</a>, <a class="ref" href="waybel_4.html#D10" target="_self">Def10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:104"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_2.html#V2">meet-continuous</a>
 ;<br/>

<b>given </b><font color="Maroon">R</font> being  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E3:104"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">R'</font> being <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> holds  <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Olive">R'</font>
 ;<br/>

<a NAME="E4:104"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Olive">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>set </b><font color="Maroon">K</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">AR</font> is  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> : <font color="Olive">AR</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">L</font> </span>}</span> ;<br/>
<a NAME="E1:104_2"/><i><font color="Green">E34</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">AR</font> is  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> : <font color="Olive">AR</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">L</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E2:104_2"/>
<font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:104_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">AR</font> is  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> : <font color="Olive">AR</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">L</font> </span>}</span> 
 
;<br/>
<a NAME="E4:104_2"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T6" target="_self">Th6</a>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E5:104_2"/><i><font color="Green">E50</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E5:104_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E6:104_2"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">AR</font> is  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> : <font color="Olive">AR</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">L</font> </span>}</span>  holds <br/><font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">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">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:104_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">AR</font> is  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> : <font color="Olive">AR</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">L</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">AA</font> being  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:104_2_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">AA</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font> &amp; <font color="Maroon">AA</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">L</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">AA</font> = <font color="Maroon">AA</font> as  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel_4.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:104_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E6:104_2_1"/>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">AA</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:104_2_1"/><b>then </b>
<font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">AA</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E3:104_2"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:104_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#T9" target="_self">Th9</a>, <a class="ref" href="waybel_4.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:104_2"/><b>then </b>
<font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">AR</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">AR</font> is  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> : <font color="Olive">AR</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K12" target="_self">App</a> <font color="Maroon">L</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<b>hence </b><a NAME="E8:104_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_4.html#T6" target="_self">Th6</a>, <a class="ref" href="waybel_4.html#T7" target="_self">Th7</a>, <a class="ref" href="waybel_4.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:104"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E6:104"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="waybel_0.html#V1">directed</a> )
 
;<br/>
<b>hence </b><a NAME="E7:104"/>
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="ref" href="waybel_4.html#T6" target="_self">Th6</a>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>


</div>
