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

<b>let </b><font color="Maroon">Q</font> be   <a href="quantal1.html#NM2">Quantale</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>, <font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/>



<b>assume </b><a NAME="E1:47"/>
<font color="Maroon">s</font> is <a href="quantal1.html#V18" target="_self">cyclic</a>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">s</font> </span>}</span> ;<br/>
<a NAME="E2:47"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Olive">b'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>, <font color="Olive">b'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : ( <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">a</font>) &amp; <font color="Olive">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">b</font>) ) </span>}</span> ,<font color="Maroon">Q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:47"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Olive">b'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>, <font color="Olive">b'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : ( <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">a</font>) &amp; <font color="Olive">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">b</font>) ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">a'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Olive">b'</font></span>)</span> where <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>, <font color="Olive">b'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : ( <font color="Olive">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">a</font>) &amp; <font color="Olive">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">b</font>) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>, <font color="Maroon">b'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">a</font>) &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">b</font>) )
 ;<br/>
<a NAME="E4:47_1"/><i><font color="Green">E28</font></i>: 
(  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> st <br/>( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">s</font> ) &amp;  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> st <br/>( <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">s</font> ) )
 
<b>by </b><i/>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>6</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Q</font> = <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b'</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>7</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>) <b>-&gt; </b>  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> = <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">s</font>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>7</sub></font>(<font color="Olive">c</font>) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">H<sub>7</sub></font>(<font color="Olive">c</font>)) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span> ;<br/>
<a NAME="E5:47_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive">c</font>) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>7</sub></font>(<font color="Olive">c</font>) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span>  </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">H<sub>7</sub></font>(<font color="Olive">c</font>)) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="quantal1.html#S1" target="_self">QUANTAL1:sch 1</a>();<br/></i>
<a NAME="E6:47_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b'</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">H<sub>7</sub></font>(<font color="Olive">c</font>)) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span> ,<font color="Maroon">Q</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:47_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">H<sub>7</sub></font>(<font color="Olive">c</font>)) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:47_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">H<sub>7</sub></font>(<font color="Olive">c</font>)) where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E3:47_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b'</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">s</font> )
 ;<br/>
<a NAME="E4:47_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="quantal1.html#E2:47"><i><font color="Green">E26</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E5:47_1_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">b</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b'</font> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:47_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b'</font> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<a NAME="E7:47_1_1"/><b>then </b>
<font color="Maroon">b'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:47_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E9:47_1_1"/><b>then </b>
( <font color="Maroon">b'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">a</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a'</font> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:47_1_1"/><b>then </b>
<font color="Maroon">b'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a'</font> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<a NAME="E11:47_1_1"/><b>then </b>
<font color="Maroon">a'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">s</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:47_1_1"/>
<font color="Maroon">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="quantal1.html#E2:47"><i><font color="Green">E26</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:47_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b'</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E9:47_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font>)
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:47"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font>),<font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T46">LATTICE3:46</a></i>;<br/>
<b>hence </b><a NAME="E5:47"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">s</font>
 ;<br/>


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