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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:127"/><span class="lab"><font color="Green" title="E77">A1</font></span>: 
<font color="Maroon" title="c1">H</font> is  <a href="waybel_1.html#V9" title="WAYBEL_1:attr.9">Heyting</a> 
 <span class="kw">and </span><br/><a NAME="E2:127"/><span class="lab"><font color="Green" title="E78">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<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">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/>
<span class="kw">set </span><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="E3:127"/><span class="lab"><font color="Green" title="E79">A3</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">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="127_1"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><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:()"><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">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"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">assume </span><a NAME="E1:127_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:127_1_1"/><span class="lab"><font color="Green" title="E80">A4</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E4:127_1_1"/><span class="lab"><font color="Green" title="E81">A5</font></span>: 
<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="E5:127_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E4:127_1_1"><span class="lab"><font color="Green" title="E81">A5</font></span></a>, <a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></span>;<br/><span class="kw">hence </span><a NAME="E6:127_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127_1_1"><span class="lab"><font color="Green" title="E80">A4</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">let </span><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E2:127_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:127_1"/><span class="lab"><font color="Green" title="E80">A6</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E5:127_1"/><span class="lab"><font color="Green" title="E81">A7</font></span>: 
<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="E6:127_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E2:127"><span class="lab"><font color="Green" title="E78">A2</font></span></a>, <a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></span>;<br/>
<a NAME="E7:127_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E5:127_1"><span class="lab"><font color="Green" title="E81">A7</font></span></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:127_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:127_1"><span class="lab"><font color="Green" title="E80">A6</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>
<div><span class="lab"><font color="Green" title="E80">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="127_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:127_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></span>;<br/><a NAME="E2:127_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="waybel_1.html#T70" target="_self" title="WAYBEL_1:th.70">Th70</a></span>;<br/><a NAME="E3:127_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A9</font></span>: 
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127"><span class="lab"><font color="Green" title="E79">A3</font></span></a></span>;<br/><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">H</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="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/><span class="kw">assume </span><a NAME="E4:127_2"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">hence </span><a NAME="E5:127_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#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="#E3:127_2"><span class="lab"><font color="Green" title="E81">A9</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="E5:127"/>
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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="waybel_1.html#D19" target="_self" title="WAYBEL_1:def.19">Def19</a></span>;<br/>
<a NAME="E6:127"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A10</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E3:127"><span class="lab"><font color="Green" title="E79">A3</font></span></a>, <a class="ref" href="waybel_1.html#T65" target="_self" title="WAYBEL_1:th.65">Th65</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:127"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#D5" target="_self" title="WAYBEL_1:def.5">Def5</a></span>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E8:127"/>
 <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="127_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">H</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> <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/>

<span class="kw">assume </span><a NAME="E1:127_3"/>
<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:()"><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>  <a href="waybel_1.html#K7" title="WAYBEL_1:func.7">'not'</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E2:127_3"/><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">H</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127"><span class="lab"><font color="Green" title="E79">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:127_3"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="waybel_1.html#T70" target="_self" title="WAYBEL_1:th.70">Th70</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="E9:127"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="txt" href="#E4:127"><span class="lab"><font color="Green" title="E80">A8</font></span></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></span>; <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#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/>

<span class="kw">thus </span><a NAME="E10:127"/>
 <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:127"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="ref" href="waybel_1.html#D5" target="_self" title="WAYBEL_1:def.5">Def5</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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