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

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  iff  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 <br/> ( <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  is  <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a>  ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">AR</font> = <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a> ;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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 <br/> ( <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  is  <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a>  ) ) implies <font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:103_1"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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 <br/> ( <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  is  <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a>  )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">L9</font> = <font color="Maroon" title="c1">L</font> as   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ;<br/><span class="kw">thus </span><a NAME="E3:103_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 <br/> ( <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  is  <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">R</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">R</font> &amp; <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  is  <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">R9</font> = <font color="Maroon" title="c4">R</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="c3">L9</font> ;<br/>
<a NAME="E2:103_1_1"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:103_1_1_1"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">a9</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b9</font> = <font color="Maroon" title="c7">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">L9</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E3:103_1_1_1"/>
<font color="Maroon" title="c8">a9</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103_1_1_1"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="ref" href="waybel_4.html#D1" target="_self" title="WAYBEL_4:def.1">Def2</a></span>;<br/>
<a NAME="E4:103_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A3</font></span>: 
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#T7" title="WAYBEL_3:th.7">WAYBEL_3:7</a></span>;<br/>
<a NAME="E5:103_1_1_1"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
 <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">A1</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c9">b9</font></span>)</span> where <font color="Olive" title="b1">A1</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="c3">L9</font> : <font color="Olive" title="b1">A1</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="c3">L9</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="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_4.html#T44" target="_self" title="WAYBEL_4:th.44">Th44</a></span>;<br/>
<a NAME="E6:103_1_1_1"/>
<font color="Maroon" title="c5">R9</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="c3">L9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_4.html#D19" target="_self" title="WAYBEL_4:def.19">Def20</a></span>;<br/>
<a NAME="E7:103_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">R9</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c9">b9</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">A1</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c9">b9</font></span>)</span> where <font color="Olive" title="b1">A1</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="c3">L9</font> : <font color="Olive" title="b1">A1</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="c3">L9</font> </span> } </span> 
 
;<br/>
<a NAME="E8:103_1_1_1"/><span class="kw">then </span>
 <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c9">b9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R9</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c9">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:103_1_1_1"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="setfam_1.html#T3" title="SETFAM_1:th.3">SETFAM_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:103_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:103_1_1_1"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="waybel_4.html#T13" target="_self" title="WAYBEL_4:th.13">Th13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:103_1_1"/>
<font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  is  <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a> </span><br/>

<span class="kw">thus </span><a NAME="E4:103_1_1"/>
<font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  is  <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103_1"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:103"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
 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 <br/> ( <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">R</font> &amp; <font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a>  is  <a href="waybel_4.html#V6" title="WAYBEL_4:attr.6">approximating</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> </span><br/>

<a NAME="E3:103"/>
 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="103_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E1:103_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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_4.html#K1" title="WAYBEL_4:func.1">-waybelow</a></span>)</span> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c3">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="waybel_4.html#D17" target="_self" title="WAYBEL_4:def.17">Def18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:103_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_4.html#T40" target="_self" title="WAYBEL_4:th.40">Th40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:103"/><span class="kw">then </span>
<font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_3.html#D5" title="WAYBEL_3:def.5">WAYBEL_3:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:103"/>
<font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
