<?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>;<br/><b>let </b><font color="Maroon" title="c2">s</font>, <font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>;<br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c2">s</font> is <a href="quantal1.html#V18" title="QUANTAL1:attr.18">cyclic</a>
 ;<br/>

<a NAME="E2:45"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</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">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</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> <span class="p1">(<span class="default"><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></span>)</span> <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="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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><br/>

<b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</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">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">b</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="E3:45_1"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> )
 ;<br/>
<a NAME="E4:45_1"/>
( <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> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">b</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">b</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#E3:45_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="quantal1.html#T12" target="_self" title="QUANTAL1:th.12">Th12</a>, <a class="ref" href="quantal1.html#T13" target="_self" title="QUANTAL1:th.13">Th13</a></i>;<br/>
<a NAME="E5:45_1"/><b>then </b>
( <font color="Maroon" title="c5">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <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#E1:45"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="quantal1.html#T8" target="_self" title="QUANTAL1:th.8">Th8</a>, <a class="ref" href="quantal1.html#T15" target="_self" title="QUANTAL1:th.15">Th15</a></i>;<br/>
<a NAME="E6:45_1"/><b>then </b>
<font color="Maroon" title="c5">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E7:45_1"/>
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><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></span>)</span> <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#E3:45_1"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:45"/><b>then </b>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</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="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> </span>}</span> ,<font color="Maroon" title="c1">Q</font> &amp;  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</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">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> </span>}</span> ,<font color="Maroon" title="c1">Q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</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> <span class="p1">(<span class="default"><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></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font> </span>}</span> ,<font color="Maroon" title="c1">Q</font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T45" title="LATTICE3:th.45">LATTICE3:45</a>, <a class="ref" href="lattice3.html#T46" title="LATTICE3:th.46">LATTICE3:46</a></i>;<br/>
<b>hence </b><a NAME="E4:45"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">s</font>
 ;<br/>

<a NAME="E5:45"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</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">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</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> : <span class="p1">(<span class="default"><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>)</span> <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="45_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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><br/>

<b>assume </b><a NAME="E1:45_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</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">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">b</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="E3:45_2"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c5">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> )
 ;<br/>
<a NAME="E4:45_2"/>
( <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> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">b</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font> &amp; <font color="Maroon" title="c5">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <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#E3:45_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="quantal1.html#T11" target="_self" title="QUANTAL1:th.11">Th11</a>, <a class="ref" href="quantal1.html#T13" target="_self" title="QUANTAL1:th.13">Th13</a></i>;<br/>
<a NAME="E5:45_2"/><b>then </b>
( <span class="p1">(<span class="default"><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>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">b</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#E1:45"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="quantal1.html#T8" target="_self" title="QUANTAL1:th.8">Th8</a>, <a class="ref" href="quantal1.html#T15" target="_self" title="QUANTAL1:th.15">Th15</a></i>;<br/>
<a NAME="E6:45_2"/><b>then </b>
<span class="p1">(<span class="default"><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>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">s</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E7:45_2"/>
<font color="Maroon" title="c4">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> : <span class="p1">(<span class="default"><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>)</span> <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#E3:45_2"><i><font color="Green" title="E22">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:45"/><b>then </b>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</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="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> </span>}</span> ,<font color="Maroon" title="c1">Q</font> &amp;  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</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">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">a</font> </span>}</span> ,<font color="Maroon" title="c1">Q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</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> : <span class="p1">(<span class="default"><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>)</span> <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> ,<font color="Maroon" title="c1">Q</font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T45" title="LATTICE3:th.45">LATTICE3:45</a>, <a class="ref" href="lattice3.html#T46" title="LATTICE3:th.46">LATTICE3:46</a></i>;<br/>
<b>hence </b><a NAME="E7:45"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><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>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">s</font>
 ;<br/>


</div>
