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

<b>let </b><font color="Maroon" title="c1">L</font> be  <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/><b>let </b><font color="Maroon" title="c2">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="c3">A</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="c2">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>consider </b><font color="Maroon" title="c4">AA</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>;<br/>
<a NAME="E2:102"/>
<font color="Maroon" title="c4">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>
 
<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="E3:102"/><b>then </b><i><font color="Green" title="E62">A1</font></i>: 
<font color="Maroon" title="c4">AA</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c2">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="c2">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="E4:102"/><i><font color="Green" title="E63">A2</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">I</font> where <font color="Olive" title="b1">I</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">I</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="c2">x</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T34" target="_self" title="WAYBEL_4:th.34">Th34</a></i>;<br/>
<a NAME="E5:102"/><i><font color="Green" title="E64">A3</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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span>)</span> where <font color="Olive" title="b1">I</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> : verum </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="c2">x</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T43" target="_self" title="WAYBEL_4:th.43">Th43</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">I1</font> being   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E7:102"/><i><font color="Green" title="E65">A4</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c5">I1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span>)</span> where <font color="Olive" title="b1">I</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E8:102"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span>)</span> where <font color="Olive" title="b1">I</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</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="E1:102_1"/>
<font color="Maroon" title="c6">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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span>)</span> where <font color="Olive" title="b1">I</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">I</font> being   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:102_1"/><i><font color="Green" title="E66">A5</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font>
 ;<br/>
<a NAME="E4:102_1"/><i><font color="Green" title="E67">A6</font></i>: 
 <a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font> is <a href="waybel_4.html#V7" title="WAYBEL_4:attr.7">approximating</a>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T37" target="_self" title="WAYBEL_4:th.37">Th37</a></i>;<br/>
<a NAME="E5:102_1"/><i><font color="Green" title="E68">A7</font></i>: 
 <a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" title="WAYBEL_4:func.8">MonSet</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T35" target="_self" title="WAYBEL_4:th.35">Th35</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">AR</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="E7:102_1"/><i><font color="Green" title="E69">A8</font></i>: 
( <font color="Maroon" title="c8">AR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="c8">AR</font> iff  ex <font color="Olive" title="b3">x'</font>, <font color="Olive" title="b4">y'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ex <font color="Olive" title="b5">s'</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b3">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b5">s'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font> &amp; <font color="Olive" title="b3">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">s'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">y'</font> ) ) ) )
 <b>by </b><i><a class="ref" href="waybel_4.html#D16" target="_self" title="WAYBEL_4:def.16">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">AR1</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><br/><a NAME="E9:102_1"/><i><font color="Green" title="E70">A9</font></i>: 
<font color="Maroon" title="c9">AR1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" title="WAYBEL_4:func.10">Map2Rel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_4.html#E4:102_1"><i><font color="Green" title="E67">A6</font></i></a>, <a class="txt" href="waybel_4.html#E5:102_1"><i><font color="Green" title="E68">A7</font></i></a>, <a class="ref" href="waybel_4.html#T42" target="_self" title="WAYBEL_4:th.42">Th42</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">ii</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E11:102_1"/><i><font color="Green" title="E71">A10</font></i>: 
( <font color="Maroon" title="c10">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">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> <font color="Maroon" title="c10">ii</font> )
 <b>by </b><i><a class="txt" href="waybel_4.html#E4:102_1"><i><font color="Green" title="E67">A6</font></i></a>, <a class="ref" href="waybel_4.html#D19" target="_self" title="WAYBEL_4:def.19">Def19</a></i>;<br/>
<a NAME="E12:102_1"/><i><font color="Green" title="E72">A11</font></i>: 
<font color="Maroon" title="c8">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:102_1_1"/>
<font color="Maroon" title="c8">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</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="E1:102_1_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<a NAME="E2:102_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">a</font>,<font color="Maroon" title="c2">x</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="c8">AR</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T13" target="_self" title="WAYBEL_4:th.13">Th13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">x'</font>, <font color="Maroon" title="c13">y'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>, <font color="Maroon" title="c14">s'</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:102_1_1_1"/><i><font color="Green" title="E73">A12</font></i>: 
( <font color="Maroon" title="c12">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> &amp; <font color="Maroon" title="c13">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c14">s'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font> &amp; <font color="Maroon" title="c12">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">s'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">y'</font> )
 <b>by </b><i><a class="txt" href="waybel_4.html#E7:102_1"><i><font color="Green" title="E69">A8</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:102_1_1_1"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E4:102_1_1_1"><i><font color="Green" title="E73">A12</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:102_1_1"/>
<span class="p1">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c2">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</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="E1:102_1_1_2"/><i><font color="Green" title="E73">A13</font></i>: 
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" title="WAYBEL_4:func.11">DownMap</a> <font color="Maroon" title="c7">I</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">a'</font> = <font color="Maroon" title="c11">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel_4.html#E11:102_1"><i><font color="Green" title="E71">A10</font></i></a></i>;<br/>
<a NAME="E3:102_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">AR</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E7:102_1"><i><font color="Green" title="E69">A8</font></i></a>, <a class="txt" href="waybel_4.html#E1:102_1_1_2"><i><font color="Green" title="E73">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:102_1_1_2"/>
<font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c2">x</font>
 ;<br/>


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


</div><b>end;</b></div>
<a NAME="E13:102_1"/>
<font color="Maroon" title="c8">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>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E7:102_1"><i><font color="Green" title="E69">A8</font></i></a>, <a class="txt" href="waybel_4.html#E9:102_1"><i><font color="Green" title="E70">A9</font></i></a>, <a class="ref" href="waybel_4.html#D20" target="_self" title="WAYBEL_4:def.20">Def20</a></i>;<br/>
<b>hence </b><a NAME="E14:102_1"/>
<font color="Maroon" title="c6">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="c2">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#E3:102_1"><i><font color="Green" title="E66">A5</font></i></a>, <a class="txt" href="waybel_4.html#E12:102_1"><i><font color="Green" title="E72">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:102"/>
 <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="c2">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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E5:102"><i><font color="Green" title="E64">A3</font></i></a>, <a class="txt" href="waybel_4.html#E7:102"><i><font color="Green" title="E65">A4</font></i></a>, <a class="ref" href="setfam_1.html#T7" title="SETFAM_1:th.7">SETFAM_1:7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E10:102"/>
 <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">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="c2">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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2"><b>proof </b></a><div class="add">

<a NAME="E1:102_2"/>
<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="c2">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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">I</font> where <font color="Olive" title="b1">I</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">I</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</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="E1:102_2_1"/>
<font color="Maroon" title="c6">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="c2">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="c7">AR</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="E3:102_2_1"/><i><font color="Green" title="E66">A14</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c7">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> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">AR</font> = <font color="Maroon" title="c7">AR</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#E3:102_2_1"><i><font color="Green" title="E66">A14</font></i></a>, <a class="ref" href="waybel_4.html#D20" target="_self" title="WAYBEL_4:def.20">Def20</a></i>;<br/>
<a NAME="E5:102_2_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">AR</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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/>
<b>hence </b><a NAME="E6:102_2_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">I</font> where <font color="Olive" title="b1">I</font> is   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">I</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_4.html#E3:102_2_1"><i><font color="Green" title="E66">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:102_2"/>
 <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Maroon" title="c2">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="c2">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#E3:102"><i><font color="Green" title="E62">A1</font></i></a>, <a class="txt" href="waybel_4.html#E4:102"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="setfam_1.html#T7" title="SETFAM_1:th.7">SETFAM_1:7</a></i>;<br/>


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


</div>
