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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> ) iff for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c1">L</font> )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c1">L</font> ) implies for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c1">L</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c1">L</font></span><br/><span class="kw">set </span><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span> ;<br/><a NAME="E2:30_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</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="E4:30_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E5:30_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
for <font color="Olive" title="b1">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 <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E6:30_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">c</font> &amp; ( for <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 <span class="p1">{<span class="default"> <font color="Olive" title="b3">x</font> where <font color="Olive" title="b3">x</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="b3">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">c</font> ) holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D7" title="YELLOW_0:def.7">YELLOW_0:def 7</a></span>;<br/><a NAME="E7:30_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> or <font color="Maroon" title="c4">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font></span><br/>

<a NAME="E2:30_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span>  or <font color="Maroon" title="c6">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_1_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font></span><br/>

<a NAME="E2:30_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</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="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:30_1_1_1"/>
<font color="Maroon" title="c6">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:30_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:30_1"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
for <font color="Olive" title="b1">c</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="Maroon" title="c2">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">c</font> &amp; ( for <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 <font color="Maroon" title="c2">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> ) holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">c</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c5">c</font> &amp; ( for <font color="Olive" title="b1">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 <font color="Maroon" title="c2">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font> ) implies <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_1_2"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c2">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c5">c</font>
 <span class="kw">and </span><br/><a NAME="E2:30_1_2"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
for <font color="Olive" title="b1">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 <font color="Maroon" title="c2">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><br/>

<a NAME="E3:30_1_2"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
for <font color="Olive" title="b1">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 <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">b</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="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c6">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">c</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_2_1"/><span class="lab"><font color="Green" title="E18">A11</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">c</font></span><br/>

<a NAME="E2:30_1_2_1"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:30_1_2_1"/>
<font color="Maroon" title="c6">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:30_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span>  <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span>  or <font color="Maroon" title="c6">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_2_2"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font></span><br/>

<a NAME="E2:30_1_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</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="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:30_1_2_2"/>
<font color="Maroon" title="c6">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:30_1_2"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E3:30_1_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:30_1"/>
for <font color="Olive" title="b1">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 <font color="Maroon" title="c2">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c4">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">b</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Y</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c4">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1_3"/>
<font color="Maroon" title="c5">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">b</font></span><br/>

<a NAME="E2:30_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</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">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c2">Y</font> </span>}</span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:30_1_3"/>
<font color="Maroon" title="c4">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:30_1"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E8:30_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="yellow_0.html#D8" title="YELLOW_0:def.8">YELLOW_0:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:30"/><span class="lab"><font color="Green" title="E9">A12</font></span>: 
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Olive" title="b1">Y</font>,<font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">Y</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span> ;<br/>
<a NAME="E3:30"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E9">A12</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</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="E5:30"/><span class="lab"><font color="Green" title="E10">A13</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E6:30"/><span class="lab"><font color="Green" title="E11">A14</font></span>: 
for <font color="Olive" title="b1">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 <span class="p1">{<span class="default"> <font color="Olive" title="b2">y</font> where <font color="Olive" title="b2">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">y</font> </span>}</span>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E7:30"/><span class="lab"><font color="Green" title="E12">A15</font></span>: 
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <span class="p1">{<span class="default"> <font color="Olive" title="b2">y</font> where <font color="Olive" title="b2">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">y</font> </span>}</span>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">c</font> &amp; ( for <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 <span class="p1">{<span class="default"> <font color="Olive" title="b3">y</font> where <font color="Olive" title="b3">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b3">y</font> </span>}</span>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">c</font> ) holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D8" title="YELLOW_0:def.8">YELLOW_0:def 8</a></span>;<br/>
<a NAME="E8:30"/><span class="lab"><font color="Green" title="E13">A16</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c4">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> or <font color="Maroon" title="c5">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_2"/><span class="lab"><font color="Green" title="E14">A17</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font></span><br/>

<a NAME="E2:30_2"/>
<font color="Maroon" title="c5">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span>  or <font color="Maroon" title="c5">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_2_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c</font></span><br/>

<a NAME="E2:30_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">y</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="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:30_2_1"/>
<font color="Maroon" title="c5">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:30_2"/>
<font color="Maroon" title="c5">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E11">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:30"/><span class="lab"><font color="Green" title="E14">A18</font></span>: 
for <font color="Olive" title="b1">c</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="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">c</font> &amp; ( for <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 <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">c</font> ) holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">c</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">c</font> &amp; ( for <font color="Olive" title="b1">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 <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">c</font> ) implies <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_3"/><span class="lab"><font color="Green" title="E15">A19</font></span>: 
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">c</font>
 <span class="kw">and </span><br/><a NAME="E2:30_3"/><span class="lab"><font color="Green" title="E16">A20</font></span>: 
for <font color="Olive" title="b1">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 <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font></span><br/>

<a NAME="E3:30_3"/><span class="lab"><font color="Green" title="E17">A21</font></span>: 
for <font color="Olive" title="b1">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 <span class="p1">{<span class="default"> <font color="Olive" title="b2">y</font> where <font color="Olive" title="b2">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">y</font> </span>}</span>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">b</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="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c6">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_3_1"/><span class="lab"><font color="Green" title="E18">A22</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font></span><br/>

<a NAME="E2:30_3_1"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3"><span class="lab"><font color="Green" title="E15">A19</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:30_3_1"/>
<font color="Maroon" title="c6">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_3_1"><span class="lab"><font color="Green" title="E18">A22</font></span></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:30_3"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span>  <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c5">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span>  or <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_3_2"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font></span><br/>

<a NAME="E2:30_3_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</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="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c2">X</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:30_3_2"/>
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_3"><span class="lab"><font color="Green" title="E16">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:30_3"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E12">A15</font></span></a>, <a class="txt" href="#E3:30_3"><span class="lab"><font color="Green" title="E17">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:30"/>
for <font color="Olive" title="b1">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 <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c4">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">b</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">b</font> implies <font color="Maroon" title="c4">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_4"/>
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font></span><br/>

<a NAME="E2:30_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</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" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">y</font> </span>}</span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:30_4"/>
<font color="Maroon" title="c4">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E10">A13</font></span></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:30"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="ref" href="yellow_0.html#D7" title="YELLOW_0:def.7">YELLOW_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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