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

<b>let </b><font color="Maroon" title="c1">H</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">H</font> is <a href="waybel_1.html#V9" title="WAYBEL_1:attr.9">Heyting</a> &amp; <font color="Maroon" title="c1">H</font> is <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> implies for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font> holds   <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Olive" title="b1">a</font> <a href="waybel_1.html#R4" title="WAYBEL_1:pred.4">is_maximum_of</a> <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">H</font> : <font color="Olive" title="b1">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span>  )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:125"/><i><font color="Green" title="E77">A1</font></i>: 
<font color="Maroon" title="c1">H</font> is <a href="waybel_1.html#V9" title="WAYBEL_1:attr.9">Heyting</a>
 <b>and </b><br/><a NAME="E2:125"/><i><font color="Green" title="E78">A2</font></i>: 
<font color="Maroon" title="c1">H</font> is <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font> holds   <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Olive" title="b1">a</font> <a href="waybel_1.html#R4" title="WAYBEL_1:pred.4">is_maximum_of</a> <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">H</font> : <font color="Olive" title="b1">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> </span><br/>

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</font> <a href="waybel_1.html#R4" title="WAYBEL_1:pred.4">is_maximum_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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> </span><br/>

<a NAME="E3:125"/><i><font color="Green" title="E79">A3</font></i>: 
<font color="Maroon" title="c1">H</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
<b>by </b><i><a class="txt" href="#E1:125"><i><font color="Green" title="E77">A1</font></i></a></i>;<br/>
<b>set </b><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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c4">Y</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> ;<br/>
<a NAME="E4:125"/><i><font color="Green" title="E80">A4</font></i>: 
<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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <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="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> </span>
<div class="add"><b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">y</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span>  implies <font color="Maroon" title="c5">y</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span>  )</span><br/><b>assume </b><a NAME="E1:125_1_1"/>
<font color="Maroon" title="c5">y</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</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="c5">y</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> </span><br/><b>then consider </b><font color="Maroon" title="c6">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font><b> such that </b><br/><a NAME="E3:125_1_1"/><i><font color="Green" title="E81">A5</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>and </b><br/><a NAME="E4:125_1_1"/><i><font color="Green" title="E82">A6</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font>
 ;<br/><a NAME="E5:125_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E3:125"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="#E4:125_1_1"><i><font color="Green" title="E82">A6</font></i></a>, <a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></i>;<br/><b>hence </b><a NAME="E6:125_1_1"/>
<font color="Maroon" title="c5">y</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E3:125_1_1"><i><font color="Green" title="E81">A5</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>

<b>let </b><font color="Maroon" title="c5">y</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="c5">y</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span>  or <font color="Maroon" title="c5">y</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E2:125_1"/>
<font color="Maroon" title="c5">y</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</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="c5">y</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c6">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font><b> such that </b><br/><a NAME="E4:125_1"/><i><font color="Green" title="E81">A7</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>and </b><br/><a NAME="E5:125_1"/><i><font color="Green" title="E82">A8</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font>
 ;<br/>
<a NAME="E6:125_1"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="#E2:125"><i><font color="Green" title="E78">A2</font></i></a>, <a class="txt" href="#E3:125"><i><font color="Green" title="E79">A3</font></i></a>, <a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/>
<a NAME="E7:125_1"/><b>then </b>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="#E3:125"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="#E5:125_1"><i><font color="Green" title="E82">A8</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/>
<b>hence </b><a NAME="E8:125_1"/>
<font color="Maroon" title="c5">y</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:125_1"><i><font color="Green" title="E81">A7</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>
<a NAME="E5:125"/>
for <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">H</font> holds  <font color="Olive" title="b1">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is <a href="waybel_1.html#V5" title="WAYBEL_1:attr.5">lower_adjoint</a>
 
<b>by </b><i><a class="txt" href="#E1:125"><i><font color="Green" title="E77">A1</font></i></a>, <a class="ref" href="waybel_1.html#D19" target="_self" title="WAYBEL_1:def.19">Def19</a></i>;<br/>
<a NAME="E6:125"/><b>then </b><i><font color="Green" title="E81">A9</font></i>: 
 <a href="waybel_1.html#R2" title="WAYBEL_1:pred.2">ex_max_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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> ,<font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="txt" href="#E3:125"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="#E4:125"><i><font color="Green" title="E80">A4</font></i></a>, <a class="ref" href="waybel_1.html#T65" target="_self" title="WAYBEL_1:th.65">Th65</a></i>;<br/>
<b>hence </b><a NAME="E7:125"/>
 <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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> ,<font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="waybel_1.html#D5" target="_self" title="WAYBEL_1:def.5">Def5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D7" target="_self" title="WAYBEL_1:def.7">WAYBEL_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> ,<font color="Maroon" title="c1">H</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> ,<font color="Maroon" title="c1">H</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span>  )</span><br/>

<a NAME="E8:125"/><i><font color="Green" title="E82">A10</font></i>: 
 <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_2"><b>proof </b></a><div class="add">

<b>let </b><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">H</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="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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span>  or <font color="Maroon" title="c5">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<b>assume </b><a NAME="E1:125_2"/>
<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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</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="c5">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font><b> such that </b><br/><a NAME="E3:125_2"/><i><font color="Green" title="E83">A11</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 <b>and </b><br/><a NAME="E4:125_2"/><i><font color="Green" title="E84">A12</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E4:125"><i><font color="Green" title="E80">A4</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:125_2"/>
 <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="#E1:125"><i><font color="Green" title="E77">A1</font></i></a>, <a class="txt" href="#E3:125_2"><i><font color="Green" title="E83">A11</font></i></a>, <a class="txt" href="#E4:125_2"><i><font color="Green" title="E84">A12</font></i></a>, <a class="ref" href="waybel_1.html#T70" target="_self" title="WAYBEL_1:th.70">Th70</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="125_3"><b>now </b></a><div class="add"><b>let </b><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">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span>  implies  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font> )</span><br/><b>assume </b><a NAME="E1:125_3"/><i><font color="Green" title="E83">A13</font></i>: 
<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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">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="E2:125_3"/>
<font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K6" title="WAYBEL_1:func.6">=&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="waybel_1.html#K6" title="WAYBEL_1:func.6">=&gt;</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:125"><i><font color="Green" title="E79">A3</font></i></a>, <a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></i>;<br/><a NAME="E3:125_3"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E1:125"><i><font color="Green" title="E77">A1</font></i></a>, <a class="ref" href="waybel_1.html#T70" target="_self" title="WAYBEL_1:th.70">Th70</a></i>;<br/><a NAME="E4:125_3"/><b>then </b>
 <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:125"><i><font color="Green" title="E80">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E5:125_3"/>
 <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="#E1:125_3"><i><font color="Green" title="E83">A13</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</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="E10:125"/>
 <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> ,<font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E3:125"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="#E8:125"><i><font color="Green" title="E82">A10</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> ,<font color="Maroon" title="c1">H</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> </span><br/>

<b>thus </b><a NAME="E11:125"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> ,<font color="Maroon" title="c1">H</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">H</font> : <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">H</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E6:125"><i><font color="Green" title="E81">A9</font></i></a>, <a class="ref" href="waybel_1.html#D5" target="_self" title="WAYBEL_1:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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