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

<b>let </b><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#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> or <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V25" title="WAYBEL_0:attr.25">/\-complete</a> ) implies for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <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 <font color="Olive" title="b1">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b3">z</font> holds <br/>(  <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>,<font color="Maroon" title="c1">L</font> &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="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b3">z</font> ) )</span><br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> or <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V25" title="WAYBEL_0:attr.25">/\-complete</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <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 <font color="Olive" title="b1">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b3">z</font> holds <br/>(  <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>,<font color="Maroon" title="c1">L</font> &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="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b3">z</font> )</span><br/>

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c3">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">z</font> implies (  <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="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">z</font> ) )</span><br/>

<b>assume </b><a NAME="E2:7"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="waybel_3.html#NR3" title="WAYBEL_3:NR.3">&gt;&gt;</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">z</font> or (  <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="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">z</font> ) )</span><br/>

<a NAME="E3:7"/><b>then </b><i><font color="Green" title="E8">A3</font></i>: 
( <font color="Maroon" title="c4">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">x</font> &amp; ( for <font color="Olive" title="b1">D</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">D</font> holds <br/> ex <font color="Olive" title="b2">d</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">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp; <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">d</font> ) ) )
 
<b>by </b><i><a class="ref" href="waybel_3.html#D1" target="_self" title="WAYBEL_3:def.1">Def1</a>, <a class="ref" href="waybel_3.html#T1" target="_self" title="WAYBEL_3:th.1">Th1</a></i>;<br/>
<b>assume </b><a NAME="E4:7"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="waybel_3.html#NR3" title="WAYBEL_3:NR.3">&gt;&gt;</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">z</font> )</span><br/>

<a NAME="E5:7"/><b>then </b><i><font color="Green" title="E10">A5</font></i>: 
( <font color="Maroon" title="c4">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">y</font> &amp; ( for <font color="Olive" title="b1">D</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Olive" title="b1">D</font> holds <br/> ex <font color="Olive" title="b2">d</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">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">d</font> ) ) )
 
<b>by </b><i><a class="ref" href="waybel_3.html#D1" target="_self" title="WAYBEL_3:def.1">Def1</a>, <a class="ref" href="waybel_3.html#T1" target="_self" title="WAYBEL_3:th.1">Th1</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">z</font></span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1"/>
( <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> or <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V25" title="WAYBEL_0:attr.25">/\-complete</a> )
 </span><b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">L</font></span><br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1"/>
 <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="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T20" title="YELLOW_0:th.20">YELLOW_0:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_2"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V25" title="WAYBEL_0:attr.25">/\-complete</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">L</font></span><br/><div class="add"><b>set </b><font color="Maroon" title="c5">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</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">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">y</font> ) </span>}</span> ;<br/><a NAME="E2:7_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</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">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">y</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</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">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">y</font> ) </span>}</span>  or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>assume </b><a NAME="E1:7_1_1_2_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</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">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">y</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:7_1_1_2_1"/><b>then </b>
 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="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">y</font> )
 
;<br/>
<b>hence </b><a NAME="E3:7_1_1_2_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c6">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</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">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">y</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> ;<br/><a NAME="E4:7_1_1_2"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="#E5:7"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/><a NAME="E5:7_1_1_2"/><b>then </b>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E1:7_1_1_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="waybel_0.html#T76" title="WAYBEL_0:th.76">WAYBEL_0:76</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E7:7_1_1_2"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c6">X</font>
 <b>and </b><br/><a NAME="E8:7_1_1_2"/><i><font color="Green" title="E14">A9</font></i>: 
for <font color="Olive" title="b1">d</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">d</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c6">X</font> holds <br/><font color="Olive" title="b1">d</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T16" title="YELLOW_0:th.16">YELLOW_0:16</a></i>;<br/><a NAME="E9:7_1_1_2"/><i><font color="Green" title="E15">A10</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font> )</span><br/>

<b>assume </b><a NAME="E1:7_1_1_2_2"/><i><font color="Green" title="E16">A11</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font></span><br/>

<a NAME="E2:7_1_1_2_2"/>
<font color="Maroon" title="c8">a</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c6">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font> or <font color="Maroon" title="c8">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font> )</span><br/>

<b>assume </b><a NAME="E1:7_1_1_2_2_1"/>
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font></span><br/>

<a NAME="E2:7_1_1_2_2_1"/><b>then </b>
 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="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">y</font> )
 
;<br/>
<b>hence </b><a NAME="E3:7_1_1_2_2_1"/>
<font color="Maroon" title="c8">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="#E1:7_1_1_2_2"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_1_2_2"/>
<font color="Maroon" title="c8">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="#E8:7_1_1_2"><i><font color="Green" title="E14">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">a</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">a</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies <font color="Maroon" title="c7">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">a</font> )</span><br/><b>assume </b><a NAME="E1:7_1_1_2_3"/>
<font color="Maroon" title="c8">a</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">a</font></span><br/><a NAME="E2:7_1_1_2_3"/><b>then </b>
( <font color="Maroon" title="c8">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c8">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T8" title="YELLOW_0:th.8">YELLOW_0:8</a></i>;<br/><a NAME="E3:7_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 ;<br/><b>hence </b><a NAME="E4:7_1_1_2_3"/>
<font color="Maroon" title="c7">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="txt" href="#E7:7_1_1_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E11:7_1_1_2"/>
 <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="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E9:7_1_1_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="yellow_0.html#T15" title="YELLOW_0:th.15">YELLOW_0:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D1" target="_self" title="WAYBEL_3:def.1">WAYBEL_3:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font> implies  ex <font color="Olive" title="b1">d</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">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font> ) )</span><br/>

<b>assume </b><a NAME="E7:7"/><i><font color="Green" title="E12">A12</font></i>: 
<font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">d</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">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">d</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c6">d1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E9:7"/><i><font color="Green" title="E13">A13</font></i>: 
( <font color="Maroon" title="c6">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">d1</font> )
 <b>by </b><i><a class="txt" href="#E2:7"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="waybel_3.html#D1" target="_self" title="WAYBEL_3:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">d2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E11:7"/><i><font color="Green" title="E14">A14</font></i>: 
( <font color="Maroon" title="c7">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">d2</font> )
 <b>by </b><i><a class="txt" href="#E4:7"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="#E7:7"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="waybel_3.html#D1" target="_self" title="WAYBEL_3:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E13:7"/><i><font color="Green" title="E15">A15</font></i>: 
( <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c6">d1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c7">d2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font> )
 <b>by </b><i><a class="txt" href="#E9:7"><i><font color="Green" title="E13">A13</font></i></a>, <a class="txt" href="#E11:7"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></i>;<br/>
<a NAME="E14:7"/><i><font color="Green" title="E16">A16</font></i>: 
( <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font> )
 
<b>by </b><i><a class="txt" href="#E9:7"><i><font color="Green" title="E13">A13</font></i></a>, <a class="txt" href="#E11:7"><i><font color="Green" title="E14">A14</font></i></a>, <a class="txt" href="#E13:7"><i><font color="Green" title="E15">A15</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font> )</span><br/>

<b>thus </b><a NAME="E15:7"/>
( <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">d</font> )
 <b>by </b><i><a class="txt" href="#E6:7"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="#E14:7"><i><font color="Green" title="E16">A16</font></i></a>, <a class="ref" href="yellow_0.html#T18" title="YELLOW_0:th.18">YELLOW_0:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
