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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V1">with_suprema</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>set </b><font color="Maroon">P</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) );<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>take </b>
<font color="Maroon">Z</font>
;<br/>

<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">x'</font>;<br/>
<a NAME="E4:53"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>set </b><font color="Maroon">DZ</font> =  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">Z</font>;<br/>
<a NAME="E6:53"/>
for <font color="Olive">u</font>, <font color="Olive">v</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> holds <br/> ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:53_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E2:53_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:53_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E5:53_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) )
 <b>by </b><i><a class="txt" href="yellow_2.html#E2:53_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:53_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E8:53_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) )
 <b>by </b><i><a class="txt" href="yellow_2.html#E4:53_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E9:53_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) holds <br/> ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Olive">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:53_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:53_1_1"/><i><font color="Green">E80</font></i>: 
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> )
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:53_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:53_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E6:53_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b><font color="Maroon">z</font> = <font color="Maroon">c</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>;<br/>

<a NAME="E8:53_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T5" target="_self">Th5</a>, , <a class="ref" href="waybel_0.html#T40">WAYBEL_0:40</a></i>;<br/>
<b>hence </b><a NAME="E9:53_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/>

<a NAME="E10:53_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E11:53_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E12:53_1_1"/>
( <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:53_1"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) holds <br/> ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Olive">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:53_1_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E2:53_1_2"/><i><font color="Green">E89</font></i>: 
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> )
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:53_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:53_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E6:53_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b><font color="Maroon">z</font> = <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c</font>;<br/>

<a NAME="E8:53_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y'</font>
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#T40">WAYBEL_0:40</a></i>;<br/>
<b>hence </b><a NAME="E9:53_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/>

<a NAME="E10:53_1_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E11:53_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E12:53_1_2"/>
( <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="ref" href="yellow_2.html#T7" target="_self">Th7</a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_3"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) ) or ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) ) or (  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) ) )
 </span><b>by </b><i><a class="txt" href="yellow_2.html#E7:53_1"><i><font color="Green">E29</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3_1"><b>suppose </b></a><a NAME="E1:53_1_3_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:53_1_3_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">z</font>
;<br/><b>thus </b><a NAME="E4:53_1_3_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E5:53_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow_2.html#E8:53_1"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3_2"><b>suppose </b></a><a NAME="E1:53_1_3_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">q</font>
;<br/><b>thus </b><a NAME="E2:53_1_3_2"/>
( <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">q</font> &amp; <font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E5:53_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow_2.html#E8:53_1"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3_3"><b>suppose </b></a><a NAME="E1:53_1_3_3"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:53_1_3_3"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E9:53_1"><i><font color="Green">E32</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">z</font>
;<br/><b>thus </b><a NAME="E4:53_1_3_3"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E5:53_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow_2.html#E8:53_1"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3_4"><b>suppose </b></a><a NAME="E1:53_1_3_4"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">q</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">p</font>
;<br/><b>thus </b><a NAME="E2:53_1_3_4"/>
( <font color="Maroon">q</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">p</font> &amp; <font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E5:53_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow_2.html#E8:53_1"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3_5"><b>suppose </b></a><a NAME="E1:53_1_3_5"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:53_1_3_5"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">z</font>
;<br/><b>thus </b><a NAME="E4:53_1_3_5"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E5:53_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow_2.html#E8:53_1"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3_6"><b>suppose </b></a><a NAME="E1:53_1_3_6"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:53_1_3_6"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="yellow_2.html#T5" target="_self">Th5</a></i>;<br/><b>take </b>
<font color="Maroon">z</font>
;<br/><b>thus </b><a NAME="E4:53_1_3_6"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E5:53_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow_2.html#E8:53_1"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3_7"><b>suppose </b></a><a NAME="E1:53_1_3_7"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:53_1_3_7"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E9:53_1"><i><font color="Green">E32</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">z</font>
;<br/><b>thus </b><a NAME="E4:53_1_3_7"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E5:53_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow_2.html#E8:53_1"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3_8"><b>suppose </b></a><a NAME="E1:53_1_3_8"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:53_1_3_8"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="yellow_2.html#T5" target="_self">Th5</a></i>;<br/><b>take </b>
<font color="Maroon">z</font>
;<br/><b>thus </b><a NAME="E4:53_1_3_8"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E5:53_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow_2.html#E8:53_1"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_3_9"><b>suppose </b></a><a NAME="E1:53_1_3_9"/>
(  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:53_1_3_9"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> )
 <b>and </b><br/><a NAME="E4:53_1_3_9"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">d</font> )
 ;<br/><b>reconsider </b><font color="Maroon">ac</font> = <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c</font>, <font color="Maroon">bd</font> = <font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">d</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><b>take </b><font color="Maroon">z</font> = <font color="Maroon">ac</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">bd</font>;<br/><a NAME="E6:53_1_3_9"/>
( <font color="Maroon">ac</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">bd</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y'</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T40">WAYBEL_0:40</a></i>;<br/><b>hence </b><a NAME="E7:53_1_3_9"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E8:53_1_3_9"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">ac</font> &amp; <font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">ac</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">bd</font> &amp; <font color="Maroon">d</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">bd</font> &amp; <font color="Maroon">ac</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">bd</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E9:53_1_3_9"/><b>then </b>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E10:53_1_3_9"/>
<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="yellow_2.html#E5:53_1"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E11:53_1_3_9"/>
( <font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">d</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E12:53_1_3_9"/>
<font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="yellow_2.html#E8:53_1"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:53"/><b>then </b>
<font color="Maroon">Z</font> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<a NAME="E8:53"/><b>then </b>
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">Z</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T30">WAYBEL_0:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">DZ</font> =  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">Z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E10:53"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_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="E1:53_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E3:53_2"/>
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DZ</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a></i>;<br/>
<b>hence </b><a NAME="E4:53_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DZ</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:53"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">DZ</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E12:53"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_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="E1:53_3"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E3:53_3"/>
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">DZ</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a></i>;<br/>
<b>hence </b><a NAME="E4:53_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DZ</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:53"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">DZ</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E14:53"/>
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>  st <font color="Olive">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Olive">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">DZ</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:53_4"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font> )
 ;<br/>

<a NAME="E2:53_4"/><b>then </b><i><font color="Green">E144</font></i>: 
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">d</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E3:53_4"/>
<font color="Maroon">DZ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="E1:53_4_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DZ</font>
 ;<br/>

<a NAME="E2:53_4_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#T14">WAYBEL_0:14</a></i>;<br/>
<b>then consider </b><font color="Maroon">p'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:53_4_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E5:53_4_1"/><i><font color="Green">E147</font></i>: 
 ex <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">p'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 ;<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="E7:53_4_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">p'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E8:53_4_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E10:53_4_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E11:53_4_1"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_4_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1_1"><b>suppose </b></a><a NAME="E1:53_4_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:53_4_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font>
 <b>by </b><i>, , <a class="ref" href="yellow_2.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/><b>hence </b><a NAME="E3:53_4_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1_2"><b>suppose </b></a><a NAME="E1:53_4_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:53_4_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="yellow_2.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/><b>hence </b><a NAME="E3:53_4_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_4_1_1_3"><b>suppose </b></a><a NAME="E1:53_4_1_1_3"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:53_4_1_1_3"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:53_4_1_1_3"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E5:53_4_1_1_3"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 ;<br/><b>reconsider </b><font color="Maroon">d'</font> = <font color="Maroon">d</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><a NAME="E7:53_4_1_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d'</font>
 <b>by </b><i>, <a class="ref" href="yellow_2.html#T5" target="_self">Th5</a>, , , , <a class="ref" href="waybel_0.html#T40">WAYBEL_0:40</a></i>;<br/><b>hence </b><a NAME="E8:53_4_1_1_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:53_4"/>
<font color="Maroon">DZ</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:53"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> or  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b</font> ) ) </span>}</span>  &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T18">YELLOW_0:18</a></i>;<br/>


</div>
