<?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>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">R</font> be  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="waybel_4.html#V6" target="_self">approximating</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>;<br/>







<b>assume </b><a NAME="E1:113"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 ;<br/>

<b>set </b><font color="Maroon">I</font> = <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">u</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/>
<a NAME="E2:113"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:113"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:113"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">u</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_4.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:113"/>
<span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">u</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</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="E1:113_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">u</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:113_1"/><b>then </b>
 ex <font color="Olive">u1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">u1</font> &amp;  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">u1</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:113_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">I</font> = <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">u</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel_4.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E7:113"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">I</font> is <a href="waybel_0.html#V12">lower</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a><br/>

<b>assume </b><a NAME="E1:113_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">y1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:113_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp;  ex <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Olive">s1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">s1</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">s1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:113_2"/><i><font color="Green">E50</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:113_2"/>
<font color="Maroon">s1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s1</font>
 
;<br/>
<a NAME="E7:113_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">s1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E8:113_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E8:113"/>
<font color="Maroon">I</font> is <a href="waybel_0.html#V1">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">u2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E1:113_3"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">v1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:113_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> &amp;  ex <font color="Olive">y1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Olive">y1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y1</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">v2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:113_3"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> &amp;  ex <font color="Olive">y2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v2</font>,<font color="Olive">y2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y2</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:113_3"/><i><font color="Green">E101</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">y1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="txt" href="waybel_4.html#E1:113_3"><i><font color="Green">E74</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">y2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:113_3"/><i><font color="Green">E102</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v2</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="txt" href="waybel_4.html#E3:113_3"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">u1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u2</font>
;<br/>

<a NAME="E10:113_3"/><i><font color="Green">E103</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y2</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E1:113_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="waybel_4.html#E3:113_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="waybel_4.html#E5:113_3"><i><font color="Green">E76</font></i></a>, , </i>;<br/>
<a NAME="E11:113_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y2</font></span>)</span>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E5:113_3"><i><font color="Green">E76</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E12:113_3"/>
( <font color="Maroon">u1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">u1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">u1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u2</font> &amp; <font color="Maroon">u2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">u1</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u2</font> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">I</font> = <font color="Maroon">I</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E10:113"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">z'</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font>;<br/>
<b>assume </b><a NAME="E1:113_4"/><i><font color="Green">E104</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:113_4"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_4_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="E1:113_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:113_4_1"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp;  ex <font color="Olive">y2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Olive">y2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y2</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">y2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:113_4_1"/><i><font color="Green">E107</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y2</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:113_4_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:113_4_1"/>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 
;<br/>
<a NAME="E8:113_4_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:113_4_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:113_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">z</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:113_4"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">I</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">z</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="E4:113_4"/><b>then </b><i><font color="Green">E154</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</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"><font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E5:113_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:113_4"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E7:113_4"/><b>then </b>
not <font color="Maroon">z</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="orders_2.html#T30">ORDERS_2:30</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:113_4"/><i><font color="Green">E155</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; not <font color="Maroon">y</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/>;<br/>
<b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:113_4"/><i><font color="Green">E157</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; not <font color="Maroon">u</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>, </i>;<br/>
<a NAME="E12:113_4"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:113_4"/>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">I</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">I</font>,<font color="Maroon">L</font> iff (  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">I</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">I</font> holds <br/> <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> ) ) )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<b>hence </b><a NAME="E14:113_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:113"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#T9" target="_self">Th9</a>, <a class="ref" href="waybel_3.html#T20">WAYBEL_3:20</a></i>;<br/>
<b>then consider </b><font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E13:113"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">y'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Olive">y'</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y'</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">y'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E15:113"/><i><font color="Green">E160</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">y'</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y'</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>
<a NAME="E16:113"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E17:113"/>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 
;<br/>
<a NAME="E18:113"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">y''</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E20:113"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y''</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y''</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y''</font> )
 <b>by </b><i><a class="ref" href="waybel_4.html#T9" target="_self">Th9</a>, <a class="ref" href="waybel_4.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E21:113"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">y''</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">y'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y''</font>;<br/>
<a NAME="E22:113"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y''</font> &amp; <font color="Maroon">y''</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y''</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E23:113"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y''</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T32">ORDERS_2:32</a></i>;<br/>
<a NAME="E24:113"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
;<br/>
<a NAME="E25:113"/><b>then </b><i><font color="Green">E228</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y''</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E26:113"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y''</font></span>)</span>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E27:113"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> )
 <b>by </b><i><a class="ref" href="waybel_4.html#T12" target="_self">Th12</a>, </i>;<br/>


</div>
