<?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>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">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">A1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> ;<br/>
<a NAME="E1:101"/><i><font color="Green">E31</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Olive">I</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:101_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Olive">I</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_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> ; <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="E1:101_1_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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">I1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:101_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I1</font> )
 ;<br/>
<a NAME="E4:101_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I1</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D10" target="_self">Def10</a>, </i>;<br/>
<b>hence </b><a NAME="E5:101_1_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"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Olive">I</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_4.html#D10" target="_self">Def10</a></i>;<br/>


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

<b>thus </b><a NAME="E2:101_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Olive">I</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:101_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Olive">I</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">I1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:101_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I1</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I1</font> )
 ;<br/>
<a NAME="E4:101_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D10" target="_self">Def10</a>, </i>;<br/>
<b>hence </b><a NAME="E5:101_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_4.html#D10" target="_self">Def10</a></i>;<br/>


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


</div><b>end;</b></div>
<b>set </b><font color="Maroon">A2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> ;<br/>
<a NAME="E2:101"/><i><font color="Green">E34</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:101_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_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> ; <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="E1:101_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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">I1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:101_2_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I1</font> )
 ;<br/>
<a NAME="E4:101_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>hence </b><a NAME="E5:101_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"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_4.html#T4" target="_self">Th4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:101_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:101_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">I1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:101_2_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I1</font> )
 ;<br/>
<a NAME="E4:101_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>hence </b><a NAME="E5:101_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_4.html#T4" target="_self">Th4</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E3:101"/><i><font color="Green">E50</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span>  <a href="hidden.html#R1">=</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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="xboole_0.html#K2">\/</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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:101_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="xboole_0.html#K2">\/</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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:101_3_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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">I2</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:101_3_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/>
<a NAME="E4:101_3_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I2</font> or  not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I2</font> )
 
;<br/>
<a NAME="E5:101_3_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  or <font color="Maroon">a</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:101_3_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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="xboole_0.html#K2">\/</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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E2:101_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="xboole_0.html#K2">\/</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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:101_3_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="xboole_0.html#K2">\/</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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/>

<a NAME="E2:101_3_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  or <font color="Maroon">a</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">I2</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>, <font color="Maroon">I3</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:101_3_2"/><i><font color="Green">E75</font></i>: 
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I2</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I3</font> ) )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_3_2_1"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I2</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I3</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_1"><b>suppose </b></a><a NAME="E1:101_3_2_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:101_3_2_1_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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2_1_2"><b>suppose </b></a><a NAME="E1:101_3_2_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Maroon">I3</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I3</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:101_3_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> or  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_4_1"><b>suppose </b></a><a NAME="E1:101_4_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 ;<br/><div class="add"><a NAME="E2:101_4_1"/><i><font color="Green">E101</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_4_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">A2'</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A2'</font>;<br/>
<a NAME="E4:101_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2'</font>
 
;<br/>
<b>then consider </b><font color="Maroon">I1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:101_4_1_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I1</font> )
 <b>by </b><i><a class="ref" href="waybel_4.html#D10" target="_self">Def10</a></i>;<br/>
<b>thus </b><a NAME="E7:101_4_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">Dx</font> =  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>;<br/><a NAME="E3:101_4_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><a NAME="E4:101_4_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E5:101_4_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">K</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:101_4_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">H</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:101_4_1_2"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">H</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel_4.html#T7" target="_self">Th7</a>, <a class="ref" href="yellow_2.html#T42">YELLOW_2:42</a></i>;<br/>
<a NAME="E5:101_4_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">a'</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/>
<b>hence </b><a NAME="E6:101_4_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">K</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div><a NAME="E6:101_4_1"/><b>then </b><i><font color="Green">E105</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">K</font> </span>}</span>  <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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T7">SETFAM_1:7</a></i>;<br/><a NAME="E7:101_4_1"/><i><font color="Green">E106</font></i>: 
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_1_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">I'</font> =  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E2:101_4_1_3"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I'</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/>
<a NAME="E3:101_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I'</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:101_4_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:101_4_1_3"/>
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E6:101_4_1_3"/>
for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  holds <br/><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Olive">Z1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:101_4_1_3_1"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Z1'</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:101_4_1_3_1"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Z1'</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z1'</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:101_4_1_3_1"/><i><font color="Green">E153</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1'</font>
 
<b>by </b><i>, <a class="txt" href="waybel_4.html#E1:101"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E5:101_4_1_3_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:101_4_1_3_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</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>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:101_4_1_3"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">E</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>;<br/><a NAME="E9:101_4_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><a NAME="E10:101_4_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">K</font> </span>}</span> 
 ;<br/><a NAME="E11:101_4_1"/>
for <font color="Olive">Z1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">K</font> </span>}</span>  holds <br/> <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Olive">Z1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:101_4_1_4"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">K</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">K1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:101_4_1_4"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">K1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z1</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">K1</font> )
 ;<br/>
<b>thus </b><a NAME="E4:101_4_1_4"/>
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font>
 <b>by </b><i>, <a class="ref" href="waybel_4.html#T8" target="_self">Th8</a>, <a class="ref" href="waybel_4.html#T10" target="_self">Th10</a>, <a class="txt" href="waybel_4.html#E1:101"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:101_4_1"/><b>then </b>
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">K</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_4.html#T9" target="_self">Th9</a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/><a NAME="E13:101_4_1"/><b>then </b>
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">K</font> where <font color="Olive">K</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">K</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_4.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E14:101_4_1"/>
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </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>, <a class="ref" href="waybel_4.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_4_2"><b>suppose </b></a><a NAME="E1:101_4_2"/><i><font color="Green">E157</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">O</font> =  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>;<br/><a NAME="E2:101_4_2"/><i><font color="Green">E158</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T34">WAYBEL_0:34</a></i>;<br/><a NAME="E3:101_4_2"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><a NAME="E4:101_4_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="orders_2.html#R2">&lt;</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/><a NAME="E5:101_4_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_4.html#T6" target="_self">Th6</a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><a NAME="E6:101_4_2"/><b>then </b><i><font color="Green">E159</font></i>: 
 <a href="waybel_0.html#K5">downarrow</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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_4.html#D10" target="_self">Def10</a></i>;<br/><b>reconsider </b><font color="Maroon">O1</font> =  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> ;<br/><a NAME="E8:101_4_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">O1</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T34">WAYBEL_0:34</a></i>;<br/><a NAME="E9:101_4_2"/>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">O1</font>
 ;<br/><a NAME="E10:101_4_2"/><b>then </b><i><font color="Green">E161</font></i>: 
(  <a href="waybel_0.html#K5">downarrow</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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  &amp;  <a href="waybel_0.html#K5">downarrow</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"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  )
 <b>by </b><i>, <a class="ref" href="waybel_4.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E11:101_4_2"/><i><font color="Green">E163</font></i>: 
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:101_4_2_1"/>
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:101_4_2_1_1"/>
<font color="Maroon">Z1</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">L1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:101_4_2_1_1"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">L1</font> )
 <b>by </b><i><a class="ref" href="waybel_4.html#D10" target="_self">Def10</a></i>;<br/><b>thus </b><a NAME="E4:101_4_2_1_1"/>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z1</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:101_4_2_1"/>
 <a href="waybel_0.html#K5">downarrow</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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:101_4_2"/><i><font color="Green">E166</font></i>: 
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel_4.html#T4" target="_self">Th4</a>, , <a class="ref" href="waybel_4.html#T8" target="_self">Th8</a>, <a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>;<br/><a NAME="E13:101_4_2"/><i><font color="Green">E167</font></i>: 
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#T8" target="_self">Th8</a>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/><a NAME="E14:101_4_2"/><b>then </b><i><font color="Green">E228</font></i>: 
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <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>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E15:101_4_2"/><i><font color="Green">E229</font></i>: 
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:101_4_2_2"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E2:101_4_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> </span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_2_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">L'</font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E2:101_4_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<a NAME="E3:101_4_2_2_1"/><b>then </b><i><font color="Green">E232</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L'</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E4:101_4_2_2_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">L'</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E5:101_4_2_2_1"/><b>then </b>
<font color="Maroon">L'</font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">L'</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E6:101_4_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">L'</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E7:101_4_2_2_1"/><b>then </b><i><font color="Green">E233</font></i>: 
<font color="Maroon">L'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 
;<br/>
<a NAME="E8:101_4_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_2_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_2_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:101_4_2_2_1_1_1"/>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">Y1'</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:101_4_2_2_1_1_1"/><i><font color="Green">E236</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1'</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Y1'</font> )
 ;<br/><a NAME="E4:101_4_2_2_1_1_1"/><i><font color="Green">E237</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y1'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E5:101_4_2_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y1'</font> <a href="hidden.html#R2">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E6:101_4_2_2_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y1'</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E7:101_4_2_2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:101_4_2_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:101_4_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel_4.html#T11" target="_self">Th11</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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


</div><b>end;</b></div><a NAME="E16:101_4_2"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> </span>)</span> <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"><span class="p3">(<span class="default"><a href="waybel_4.html#K11" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:101_4_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> </span>)</span>
 ;<br/>

<a NAME="E2:101_4_2_3"/><b>then </b><i><font color="Green">E238</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>thus </b><a NAME="E3:101_4_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_2_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_4_2_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:101_4_2_3_1_1"/>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Olive">I</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">I1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:101_4_2_3_1_1"/><i><font color="Green">E239</font></i>: 
( <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">I1</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I1</font> )
 ;<br/><a NAME="E4:101_4_2_3_1_1"/>
<font color="Maroon">I1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E5:101_4_2_3_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I1</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:101_4_2_3_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:101_4_2_3_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="waybel_4.html#T8" target="_self">Th8</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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


</div><b>end;</b></div><a NAME="E17:101_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E18:101_4_2"/><b>then </b><i><font color="Green">E240</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E19:101_4_2"/>
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T11">WAYBEL_3:11</a></i>;<br/><b>hence </b><a NAME="E20:101_4_2"/>
 <a href="setfam_1.html#K1">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" target="_self">DownMap</a> <font color="Olive">I</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </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="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
