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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="quantal1.html#NM1">UnOp</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">A</font></span>)</span>;<br/>



<b>thus </b><a NAME="E1:61"/>
( <font color="Maroon">f</font> is <a href="quantal1.html#V14">monotone</a> implies <font color="Maroon">f</font> is <a href="cohsp_1.html#V6">c=-monotone</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">f</font> is <a href="quantal1.html#V14">monotone</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="knaster.html#D3" target="_self">KNASTER:def 3</a><br/>

<b>assume </b><a NAME="E2:61_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:61_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T2">LATTICE3:2</a></i>;<br/>
<a NAME="E4:61_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="quantal1.html#D12">QUANTAL1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E5:61_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T2">LATTICE3:2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:61"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">f</font> is <a href="cohsp_1.html#V6">c=-monotone</a>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D12">QUANTAL1:def 12</a><br/>

<b>assume </b><a NAME="E3:61"/>
<font color="Maroon">p</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E4:61"/><b>then </b>
<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T2">LATTICE3:2</a></i>;<br/>
<a NAME="E5:61"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:61"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T2">LATTICE3:2</a></i>;<br/>


</div>
