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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  ex <font color="Olive" title="b3">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b4">z</font> where <font color="Olive" title="b4">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> or <font color="Olive" title="b4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">y</font> or  ex <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b6">b</font> ) ) </span> } </span>  &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b3">Z</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">P</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">Z</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">z</font> where <font color="Olive" title="b2">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> or <font color="Olive" title="b2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> or  ex <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">b</font> ) ) </span> } </span>  &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Olive" title="b1">Z</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> or $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) );<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">Z</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>] </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c5">Z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> or <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> or  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">b</font> ) ) </span> } </span>  &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c5">Z</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">x9</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c7">y9</font> = <font color="Maroon" title="c4">y</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T41" target="_self" title="YELLOW_2:th.41">Th43</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">z</font> =  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">x9</font>;<br/>
<a NAME="E3:52"/>
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Z</font>
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">Z</font> = <font color="Maroon" title="c5">Z</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">DZ</font> =  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c9">Z</font>;<br/>
<a NAME="E5:52"/>
 for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> holds <br/> ex <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Olive" title="b1">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:52_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp;  ex <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">b</font> ) holds <br/> ex <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Olive" title="b1">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) implies  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:52_1_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">and </span><br/><a NAME="E2:52_1_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E4:52_1_1"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">and </span><br/><a NAME="E5:52_1_1"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">and </span><br/><a NAME="E6:52_1_1"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c14">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">c</font> = <font color="Maroon" title="c14">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<span class="kw">take </span><font color="Maroon" title="c16">z</font> = <font color="Maroon" title="c13">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c15">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c12">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> )</span><br/>

<a NAME="E8:52_1_1"/>
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E5:52_1_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="waybel_0.html#T40" title="WAYBEL_0:th.40">WAYBEL_0:40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:52_1_1"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c12">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> )</span><br/>

<a NAME="E10:52_1_1"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c15">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/>
<a NAME="E11:52_1_1"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
( <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">c</font> &amp; <font color="Maroon" title="c13">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/>
<a NAME="E12:52_1_1"/>
<font color="Maroon" title="c14">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/>
<a NAME="E13:52_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:52_1_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:52_1_1"/>
( <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c12">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E10:52_1_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E11:52_1_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:52_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp;  ex <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">b</font> ) holds <br/> ex <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Olive" title="b1">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) implies  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:52_1_2"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
<font color="Maroon" title="c11">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">and </span><br/><a NAME="E2:52_1_2"/><span class="lab"><font color="Green" title="E27">A11</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E4:52_1_2"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">and </span><br/><a NAME="E5:52_1_2"/><span class="lab"><font color="Green" title="E29">A13</font></span>: 
<font color="Maroon" title="c14">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">and </span><br/><a NAME="E6:52_1_2"/><span class="lab"><font color="Green" title="E30">A14</font></span>: 
<font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c14">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">c</font> = <font color="Maroon" title="c13">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c11">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<span class="kw">take </span><font color="Maroon" title="c16">z</font> = <font color="Maroon" title="c15">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c14">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c12">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> )</span><br/>

<a NAME="E8:52_1_2"/>
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_2"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="txt" href="#E4:52_1_2"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="waybel_0.html#T40" title="WAYBEL_0:th.40">WAYBEL_0:40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:52_1_2"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52_1_2"><span class="lab"><font color="Green" title="E29">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c12">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> )</span><br/>

<a NAME="E10:52_1_2"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
<font color="Maroon" title="c15">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/>
<a NAME="E11:52_1_2"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
( <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">c</font> &amp; <font color="Maroon" title="c14">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/>
<a NAME="E12:52_1_2"/>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/>
<a NAME="E13:52_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:52_1_2"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:52_1_2"/>
( <font color="Maroon" title="c11">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c12">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1_2"><span class="lab"><font color="Green" title="E30">A14</font></span></a>, <a class="txt" href="#E10:52_1_2"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E11:52_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">let </span><font color="Maroon" title="c11">u</font>, <font color="Maroon" title="c12">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> implies  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:52_1"/><span class="lab"><font color="Green" title="E26">A17</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <span class="kw">and </span><br/><a NAME="E4:52_1"/><span class="lab"><font color="Green" title="E27">A18</font></span>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c13">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E6:52_1"/><span class="lab"><font color="Green" title="E28">A19</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u</font>
 <span class="kw">and </span><br/><a NAME="E7:52_1"/><span class="lab"><font color="Green" title="E29">A20</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1"><span class="lab"><font color="Green" title="E26">A17</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E9:52_1"/><span class="lab"><font color="Green" title="E30">A21</font></span>: 
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v</font>
 <span class="kw">and </span><br/><a NAME="E10:52_1"/><span class="lab"><font color="Green" title="E31">A22</font></span>: 
( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1"><span class="lab"><font color="Green" title="E27">A18</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1_3"/>
( ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> ) or ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) ) or ( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) ) or ( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) ) or (  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:52_1"><span class="lab"><font color="Green" title="E29">A20</font></span></a>, <a class="txt" href="#E10:52_1"><span class="lab"><font color="Green" title="E31">A22</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_1"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:52_1_3_1"/><span class="lab"><font color="Green" title="E32">A23</font></span>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x9</font> &amp; <font color="Maroon" title="c13">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c14">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c15">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )</span><br/><span class="kw">thus </span><a NAME="E4:52_1_3_1"/>
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E9:52_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E3:52_1_3_1"><span class="lab"><font color="Green" title="E32">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_2"/><span class="lab"><font color="Green" title="E32">A24</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c13">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c14">q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c14">q</font> )</span><br/><span class="kw">thus </span><a NAME="E2:52_1_3_2"/>
( <font color="Maroon" title="c13">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c14">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E9:52_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E1:52_1_3_2"><span class="lab"><font color="Green" title="E32">A24</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_3"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_3"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:52_1_3_3"/><span class="lab"><font color="Green" title="E32">A25</font></span>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c13">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c14">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c15">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )</span><br/><span class="kw">thus </span><a NAME="E4:52_1_3_3"/>
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E9:52_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E3:52_1_3_3"><span class="lab"><font color="Green" title="E32">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_4"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_4"/><span class="lab"><font color="Green" title="E32">A26</font></span>: 
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c14">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c13">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c13">p</font> )</span><br/><span class="kw">thus </span><a NAME="E2:52_1_3_4"/>
( <font color="Maroon" title="c14">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c13">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E9:52_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E1:52_1_3_4"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_5"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_5"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:52_1_3_5"/><span class="lab"><font color="Green" title="E32">A27</font></span>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">y9</font> &amp; <font color="Maroon" title="c13">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c14">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c15">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )</span><br/><span class="kw">thus </span><a NAME="E4:52_1_3_5"/>
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E9:52_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E3:52_1_3_5"><span class="lab"><font color="Green" title="E32">A27</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_6"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_6"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:52_1_3_6"/><span class="lab"><font color="Green" title="E32">A28</font></span>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c13">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c14">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c15">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )</span><br/><span class="kw">thus </span><a NAME="E4:52_1_3_6"/>
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E9:52_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E3:52_1_3_6"><span class="lab"><font color="Green" title="E32">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_7"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_7"/>
( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:52_1_3_7"/><span class="lab"><font color="Green" title="E32">A29</font></span>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c14">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c13">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c15">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )</span><br/><span class="kw">thus </span><a NAME="E4:52_1_3_7"/>
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E9:52_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E3:52_1_3_7"><span class="lab"><font color="Green" title="E32">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_8"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_8"/>
( <font color="Maroon" title="c14">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:52_1_3_8"/><span class="lab"><font color="Green" title="E32">A30</font></span>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c14">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c13">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c15">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )</span><br/><span class="kw">thus </span><a NAME="E4:52_1_3_8"/>
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E9:52_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E3:52_1_3_8"><span class="lab"><font color="Green" title="E32">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3_9"><span class="kw">suppose </span></a><a NAME="E1:52_1_3_9"/>
(  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) &amp;  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font>, <font color="Maroon" title="c17">c</font>, <font color="Maroon" title="c18">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:52_1_3_9"/><span class="lab"><font color="Green" title="E32">A31</font></span>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> )
 <span class="kw">and </span><br/><a NAME="E4:52_1_3_9"/><span class="lab"><font color="Green" title="E33">A32</font></span>: 
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c16">b</font>
 <span class="kw">and </span><br/><a NAME="E5:52_1_3_9"/><span class="lab"><font color="Green" title="E34">A33</font></span>: 
( <font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> )
 <span class="kw">and </span><br/><a NAME="E6:52_1_3_9"/><span class="lab"><font color="Green" title="E35">A34</font></span>: 
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">c</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c18">d</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">ac</font> = <font color="Maroon" title="c15">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c17">c</font>, <font color="Maroon" title="c20">bd</font> = <font color="Maroon" title="c16">b</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c18">d</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c21">z</font> = <font color="Maroon" title="c19">ac</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c20">bd</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> &amp; <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font> )</span><br/><a NAME="E8:52_1_3_9"/>
( <font color="Maroon" title="c19">ac</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x9</font> &amp; <font color="Maroon" title="c20">bd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1_3_9"><span class="lab"><font color="Green" title="E32">A31</font></span></a>, <a class="txt" href="#E5:52_1_3_9"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="ref" href="waybel_0.html#T40" title="WAYBEL_0:th.40">WAYBEL_0:40</a></span>;<br/><span class="kw">hence </span><a NAME="E9:52_1_3_9"/>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font> &amp; <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font> )</span><br/><a NAME="E10:52_1_3_9"/><span class="lab"><font color="Green" title="E36">A35</font></span>: 
<font color="Maroon" title="c19">ac</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E11:52_1_3_9"/><span class="lab"><font color="Green" title="E37">A36</font></span>: 
<font color="Maroon" title="c20">bd</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E12:52_1_3_9"/>
<font color="Maroon" title="c16">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">bd</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E13:52_1_3_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A37</font></span>: 
<font color="Maroon" title="c16">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:52_1_3_9"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><a NAME="E14:52_1_3_9"/>
<font color="Maroon" title="c15">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">ac</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E15:52_1_3_9"/><span class="kw">then </span>
<font color="Maroon" title="c15">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:52_1_3_9"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><span class="kw">hence </span><a NAME="E16:52_1_3_9"/>
<font color="Maroon" title="c11">u</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E4:52_1_3_9"><span class="lab"><font color="Green" title="E33">A32</font></span></a>, <a class="txt" href="#E13:52_1_3_9"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font></span><br/><a NAME="E17:52_1_3_9"/>
<font color="Maroon" title="c18">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">bd</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E18:52_1_3_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A38</font></span>: 
<font color="Maroon" title="c18">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:52_1_3_9"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><a NAME="E19:52_1_3_9"/>
<font color="Maroon" title="c17">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">ac</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>;<br/><a NAME="E20:52_1_3_9"/><span class="kw">then </span>
<font color="Maroon" title="c17">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:52_1_3_9"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="ref" href="orders_2.html#T3" title="ORDERS_2:th.3">ORDERS_2:3</a></span>;<br/><span class="kw">hence </span><a NAME="E21:52_1_3_9"/>
<font color="Maroon" title="c12">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:52_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E6:52_1_3_9"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E18:52_1_3_9"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E6:52"/><span class="kw">then </span>
<font color="Maroon" title="c9">Z</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">DZ</font> =  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c9">Z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T41" target="_self" title="YELLOW_2:th.41">Th43</a></span>;<br/>
<a NAME="E8:52"/><span class="lab"><font color="Green" title="E24">A39</font></span>: 
 for <font color="Olive" title="b1">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b1">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">y</font> holds <br/><font color="Maroon" title="c11">DZ</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c12">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">y</font> implies <font color="Maroon" title="c11">DZ</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">d</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:52_2"/><span class="lab"><font color="Green" title="E25">A40</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">d</font>
 <span class="kw">and </span><br/><a NAME="E2:52_2"/><span class="lab"><font color="Green" title="E26">A41</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">DZ</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">d</font></span><br/>

<a NAME="E3:52_2"/><span class="lab"><font color="Green" title="E27">A42</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_2"><span class="lab"><font color="Green" title="E26">A41</font></span></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<a NAME="E4:52_2"/><span class="lab"><font color="Green" title="E28">A43</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_2"><span class="lab"><font color="Green" title="E25">A40</font></span></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<a NAME="E5:52_2"/>
<font color="Maroon" title="c11">DZ</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">DZ</font> or <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:52_2_1"/>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">DZ</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d</font></span><br/>

<a NAME="E2:52_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">u</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T14" title="WAYBEL_0:th.14">WAYBEL_0:14</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">p9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E4:52_2_1"/><span class="lab"><font color="Green" title="E29">A44</font></span>: 
<font color="Maroon" title="c14">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font>
 <span class="kw">and </span><br/><a NAME="E5:52_2_1"/><span class="lab"><font color="Green" title="E30">A45</font></span>: 
 ex <font color="Olive" title="b1">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c14">p9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E7:52_2_1"/><span class="lab"><font color="Green" title="E31">A46</font></span>: 
<font color="Maroon" title="c14">p9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c15">u</font>
 <span class="kw">and </span><br/><a NAME="E8:52_2_1"/><span class="lab"><font color="Green" title="E32">A47</font></span>: 
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:52_2_1"><span class="lab"><font color="Green" title="E30">A45</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E10:52_2_1"/><span class="lab"><font color="Green" title="E33">A48</font></span>: 
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">u</font>
 <span class="kw">and </span><br/><a NAME="E11:52_2_1"/><span class="lab"><font color="Green" title="E34">A49</font></span>: 
( <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:52_2_1"><span class="lab"><font color="Green" title="E32">A47</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_2_1_1"/>
( <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> or  ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:52_2_1"><span class="lab"><font color="Green" title="E34">A49</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_2_1_1_1"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d</font></span><br/><div class="add"><a NAME="E2:52_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_2_1"><span class="lab"><font color="Green" title="E29">A44</font></span></a>, <a class="txt" href="#E7:52_2_1"><span class="lab"><font color="Green" title="E31">A46</font></span></a>, <a class="txt" href="#E10:52_2_1"><span class="lab"><font color="Green" title="E33">A48</font></span></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></span>;<br/><span class="kw">hence </span><a NAME="E3:52_2_1_1_1"/>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_2"><span class="lab"><font color="Green" title="E28">A43</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_2_1_1_2"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d</font></span><br/><div class="add"><a NAME="E2:52_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">y9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_2_1"><span class="lab"><font color="Green" title="E29">A44</font></span></a>, <a class="txt" href="#E7:52_2_1"><span class="lab"><font color="Green" title="E31">A46</font></span></a>, <a class="txt" href="#E10:52_2_1"><span class="lab"><font color="Green" title="E33">A48</font></span></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></span>;<br/><span class="kw">hence </span><a NAME="E3:52_2_1_1_2"/>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_2"><span class="lab"><font color="Green" title="E27">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:52_2_1_1_3"/><span class="lab"><font color="Green" title="E35">A50</font></span>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c16">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">d9</font> = <font color="Maroon" title="c12">d</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T41" target="_self" title="YELLOW_2:th.41">Th43</a></span>;<br/><a NAME="E3:52_2_1_1_3"/>
<font color="Maroon" title="c15">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">d9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_2"><span class="lab"><font color="Green" title="E28">A43</font></span></a>, <a class="txt" href="#E3:52_2"><span class="lab"><font color="Green" title="E27">A42</font></span></a>, <a class="txt" href="#E10:52_2_1"><span class="lab"><font color="Green" title="E33">A48</font></span></a>, <a class="txt" href="#E1:52_2_1_1_3"><span class="lab"><font color="Green" title="E35">A50</font></span></a>, <a class="ref" href="waybel_0.html#T40" title="WAYBEL_0:th.40">WAYBEL_0:40</a></span>;<br/><span class="kw">hence </span><a NAME="E4:52_2_1_1_3"/>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_2_1"><span class="lab"><font color="Green" title="E29">A44</font></span></a>, <a class="txt" href="#E7:52_2_1"><span class="lab"><font color="Green" title="E31">A46</font></span></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:52_2"/>
<font color="Maroon" title="c11">DZ</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:52"/>
<font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">DZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> or <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">DZ</font> )</span><br/>

<a NAME="E1:52_3"/><span class="lab"><font color="Green" title="E25">A51</font></span>: 
<font color="Maroon" title="c9">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">DZ</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T16" title="WAYBEL_0:th.16">WAYBEL_0:16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:52_3"/><span class="lab"><font color="Green" title="E26">A52</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">DZ</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">a9</font> = <font color="Maroon" title="c12">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T42" target="_self" title="YELLOW_2:th.42">Th44</a></span>;<br/>
<a NAME="E4:52_3"/>
<font color="Maroon" title="c13">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_3"><span class="lab"><font color="Green" title="E26">A52</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:52_3"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">DZ</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_3"><span class="lab"><font color="Green" title="E25">A51</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:52"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A53</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">DZ</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<a NAME="E11:52"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">DZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> or <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">DZ</font> )</span><br/>

<a NAME="E1:52_4"/><span class="lab"><font color="Green" title="E26">A54</font></span>: 
<font color="Maroon" title="c9">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">DZ</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_0.html#T16" title="WAYBEL_0:th.16">WAYBEL_0:16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:52_4"/><span class="lab"><font color="Green" title="E27">A55</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">DZ</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">a9</font> = <font color="Maroon" title="c12">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T42" target="_self" title="YELLOW_2:th.42">Th44</a></span>;<br/>
<a NAME="E4:52_4"/>
<font color="Maroon" title="c13">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_4"><span class="lab"><font color="Green" title="E27">A55</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:52_4"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">DZ</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_4"><span class="lab"><font color="Green" title="E26">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:52"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">DZ</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:52"/>
( <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> or <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> or  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b3">b</font> ) ) </span> } </span>  &amp;  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K3" title="WAYBEL_0:func.3">downarrow</a> <font color="Maroon" title="c5">Z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:52"><span class="lab"><font color="Green" title="E25">A53</font></span></a>, <a class="txt" href="#E8:52"><span class="lab"><font color="Green" title="E24">A39</font></span></a>, <a class="ref" href="yellow_0.html#T18" title="YELLOW_0:th.18">YELLOW_0:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
