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

<b>let </b><font color="Maroon" title="c1">Q</font> be   <a href="quantal1.html#NM2" title="QUANTAL1:NM.2">Quantale</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">s</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>  st <font color="Olive" title="b2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">b</font> holds <br/>( <font color="Olive" title="b3">b</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Olive" title="b1">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b3">b</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Olive" title="b1">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Olive" title="b1">s</font> )</span><br/><b>let </b><font color="Maroon" title="c2">s</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">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="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font> implies ( <font color="Maroon" title="c4">b</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c4">b</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> ) )</span><br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">b</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c4">b</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> )</span><br/>

<a NAME="E2:41"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Olive" title="b1">d</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</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">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Olive" title="b1">c</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Olive" title="b1">d</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> </span>}</span>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Olive" title="b1">c</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:41_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Olive" title="b1">d</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Olive" title="b1">c</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c6">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font><b> such that </b><br/><a NAME="E3:41_1"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c6">d</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> )
 ;<br/>
<a NAME="E4:41_1"/>
<font color="Maroon" title="c6">d</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">d</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="txt" href="quantal1.html#E1:41"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="quantal1.html#T8" target="_self" title="QUANTAL1:th.8">Th8</a></i>;<br/>
<a NAME="E5:41_1"/><b>then </b>
<font color="Maroon" title="c6">d</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font>
 
<b>by </b><i><a class="txt" href="quantal1.html#E3:41_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E6:41_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Olive" title="b1">c</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="quantal1.html#E3:41_1"><i><font color="Green" title="E18">A2</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>hence </b><a NAME="E3:41"/>
<font color="Maroon" title="c4">b</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T46" title="LATTICE3:th.46">LATTICE3:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">b</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span><br/>

<a NAME="E4:41"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Maroon" title="c4">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</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">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Maroon" title="c3">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b1">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Maroon" title="c4">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> </span>}</span>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Maroon" title="c3">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b1">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:41_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">d</font> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Maroon" title="c4">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Maroon" title="c3">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b1">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c6">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font><b> such that </b><br/><a NAME="E3:41_2"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c4">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> )
 ;<br/>
<a NAME="E4:41_2"/>
<font color="Maroon" title="c3">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">d</font>
 
<b>by </b><i><a class="txt" href="quantal1.html#E1:41"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="quantal1.html#T8" target="_self" title="QUANTAL1:th.8">Th8</a></i>;<br/>
<a NAME="E5:41_2"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font>
 
<b>by </b><i><a class="txt" href="quantal1.html#E3:41_2"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E6:41_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Maroon" title="c3">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b1">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="quantal1.html#E3:41_2"><i><font color="Green" title="E18">A3</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>hence </b><a NAME="E5:41"/>
<font color="Maroon" title="c4">b</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T46" title="LATTICE3:th.46">LATTICE3:46</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>
