<?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/>

<b>set </b><font color="Maroon">AR</font> = <font color="Maroon">L</font> <a href="waybel_4.html#K1" target="_self">-waybelow</a> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:103_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/><b>then reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L</font> as  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_2.html#V2">meet-continuous</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:103_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 <br/> ( <font color="Maroon">L</font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">R</font> &amp; <font color="Maroon">L</font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  is <a href="waybel_4.html#V6" target="_self">approximating</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">R</font> be  <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>;<br/>

<b>reconsider </b><font color="Maroon">R'</font> = <font color="Maroon">R</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> ;<br/>
<a NAME="E2:103_1_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:103_1_1_1"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="waybel_4.html#K1" target="_self">-waybelow</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:103_1_1_1"/>
<font color="Maroon">a'</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">b'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:103_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">b'</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/>
<a NAME="E5:103_1_1_1"/><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">A1</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">b'</font></span>)</span> where <font color="Olive">A1</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">A1</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">b'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:103_1_1_1"/>
<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="E7:103_1_1_1"/><b>then </b>
<font color="Maroon">R'</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">A1</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">b'</font></span>)</span> where <font color="Olive">A1</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">A1</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="E8:103_1_1_1"/><b>then </b>
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">b'</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">b'</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<b>hence </b><a NAME="E9:103_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:103_1_1"/>
<font color="Maroon">L</font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>

<b>thus </b><a NAME="E4:103_1_1"/>
<font color="Maroon">L</font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  is <a href="waybel_4.html#V6" target="_self">approximating</a>
 <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/>


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

<b>assume </b><a NAME="E2:103"/><i><font color="Green">E37</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 <br/> ( <font color="Maroon">L</font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  <a href="tarski.html#R1">c=</a> <font color="Olive">R</font> &amp; <font color="Maroon">L</font> <a href="waybel_4.html#K1" target="_self">-waybelow</a>  is <a href="waybel_4.html#V6" target="_self">approximating</a> )
 ;<br/>

<a NAME="E3:103"/>
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="103_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/>

<a NAME="E1:103_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"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="waybel_4.html#K1" target="_self">-waybelow</a> </span>)</span> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>hence </b><a NAME="E2:103_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="txt" href="waybel_4.html#E11"><i><font color="Green">Lemma55</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:103"/><b>then </b><i><font color="Green">E40</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="E5:103"/>
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="E6:103"/>
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>


</div>
