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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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"> for <font color="Olive" title="b1">f</font> being  <a href="quantal1.html#NM1" title="QUANTAL1:NM.1">UnOp</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">A</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> is <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a> iff <font color="Olive" title="b1">f</font> is <a href="cohsp_1.html#V6" title="COHSP_1:attr.6">c=-monotone</a> )</span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="quantal1.html#NM1" title="QUANTAL1:NM.1">UnOp</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">A</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="c2">f</font> is <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a> iff <font color="Maroon" title="c2">f</font> is <a href="cohsp_1.html#V6" title="COHSP_1:attr.6">c=-monotone</a> )</span><br/>



<b>thus </b><a NAME="E1:60"/>
( <font color="Maroon" title="c2">f</font> is <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a> implies <font color="Maroon" title="c2">f</font> is <a href="cohsp_1.html#V6" title="COHSP_1:attr.6">c=-monotone</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is <a href="cohsp_1.html#V6" title="COHSP_1:attr.6">c=-monotone</a> implies <font color="Maroon" title="c2">f</font> is <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> is <a href="cohsp_1.html#V6" title="COHSP_1:attr.6">c=-monotone</a></span><br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="knaster.html#D3" target="_self" title="KNASTER:def.3">KNASTER:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">y</font> implies <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font> )</span><br/>

<b>assume </b><a NAME="E2:60_1"/>
<font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span><br/>

<a NAME="E3:60_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T2" title="LATTICE3:th.2">LATTICE3:2</a></i>;<br/>
<a NAME="E4:60_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="txt" href="#E1:60_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="quantal1.html#D12" title="QUANTAL1:def.12">QUANTAL1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E5:60_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T2" title="LATTICE3:th.2">LATTICE3:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:60"/><i><font color="Green" title="E47">A2</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="cohsp_1.html#V6" title="COHSP_1:attr.6">c=-monotone</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> is <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a></span><br/>

<b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <font color="Maroon" title="c1">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D12" title="QUANTAL1:def.12">QUANTAL1:def 12</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="c3">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">q</font> or <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">q</font> )</span><br/>

<b>assume </b><a NAME="E3:60"/>
<font color="Maroon" title="c3">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">q</font></span><br/>

<a NAME="E4:60"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T2" title="LATTICE3:th.2">LATTICE3:2</a></i>;<br/>
<a NAME="E5:60"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="#E2:60"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="knaster.html#D3" target="_self" title="KNASTER:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E6:60"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T2" title="LATTICE3:th.2">LATTICE3:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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