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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="quantal1.html#V14">monotone</a> <a href="quantal1.html#NM1">UnOp</a> of <font color="Maroon">L</font>;<br/>



<b>set </b><font color="Maroon">H</font> = <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">fH</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font></span>)</span> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">uH</font> =  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">fuH</font> =  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font></span>)</span> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font>;<br/>
<b>take </b>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> )
 ;<br/><b>reconsider </b><font color="Maroon">fh</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><b>take </b><font color="Maroon">fh</font> = <font color="Maroon">fh</font>;<br/><b>thus </b><a NAME="E5:33_1"/>
( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">fh</font> &amp; <font color="Maroon">fh</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font></span>)</span> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span>  )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E2:33"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font></span>)</span> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T48">LATTICE3:48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">fh</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:33_2"/>
<font color="Maroon">fh</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font></span>)</span> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:33_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">fh</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> &amp; <font color="Maroon">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> )
 ;<br/><a NAME="E4:33_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E5:33_2"/><b>then </b>
<font color="Maroon">h</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/><b>hence </b><a NAME="E6:33_2"/>
<font color="Maroon">fh</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="quantal1.html#D12">QUANTAL1:def 12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:33"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font></span>)</span> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>
<a NAME="E5:33"/><b>then </b>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font></span>)</span> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E6:33"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<a NAME="E7:33"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p3">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="quantal1.html#D12">QUANTAL1:def 12</a></i>;<br/>
<a NAME="E8:33"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> 
 
;<br/>
<a NAME="E9:33"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/>
<b>hence </b><a NAME="E10:33"/>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <span class="p2">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">h</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> </span>}</span> ,<font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D4">ABIAN:def 4</a><br/>


</div>
