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<b>let </b><font color="Maroon" title="c1">Q</font> be   <a href="quantal1.html#NM5" title="QUANTAL1:NM.5">Girard-Quantale</a>;<br/><b>let </b><font color="Maroon" title="c2">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>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>thus </b><a NAME="E1:68"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Q</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c2">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"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c2">a</font></span>)</span> 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> ,<font color="Maroon" title="c1">Q</font>
 <b>by </b><i><a class="ref" href="quantal1.html#D6" target="_self" title="QUANTAL1:def.6">Def6</a></i>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Q</font> = $1 <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> <font color="Maroon" title="c2">a</font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>6</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>) <b>-&gt; </b>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> =  <a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>7</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Q</font> = <span class="p1">(<span class="default"><a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> $1</span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> <font color="Maroon" title="c2">a</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>;<br/>
<a NAME="E2:68"/><i><font color="Green" title="E29">A1</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive" title="b2">d</font>) where <font color="Olive" title="b2">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">S<sub>1</sub></font>[<font color="Olive" title="b2">d</font>] </span>}</span>  </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">H<sub>6</sub></font>(<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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="quantal1.html#S1" target="_self" title="QUANTAL1:sch.1">QUANTAL1:sch 1</a>();<br/></i>
<a NAME="E3:68"/><i><font color="Green" title="E30">A2</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>6</sub></font>(<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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>7</sub></font>(<font color="Olive" title="b2">d</font>) where <font color="Olive" title="b2">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">S<sub>1</sub></font>[<font color="Olive" title="b2">d</font>] </span>}</span>  </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" 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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="quantal1.html#S1" target="_self" title="QUANTAL1:sch.1">QUANTAL1:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>8</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>) <b>-&gt; </b>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> =  <a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> $1</span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p2">(<span class="default"><a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>9</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>) <b>-&gt; </b>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> = $1 <a href="quantal1.html#K9" title="QUANTAL1:func.9">delta</a> <font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E4:68"/><i><font color="Green" title="E31">A3</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> holds  <font color="Maroon">H<sub>8</sub></font>(<font color="Olive" title="b1">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>9</sub></font>(<font color="Olive" title="b1">b</font>)
 
;<br/>
<a NAME="E5:68"/><i><font color="Green" title="E32">A4</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>8</sub></font>(<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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>9</sub></font>(<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">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S5" title="FRAENKEL:sch.5">FRAENKEL:sch 5</a>(<a class="txt" href="quantal1.html#E4:68"><i><font color="Green" title="E31">A3</font></i></a>);<br/></i>
<b>thus </b><span class="p1">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">Q</font></span>)</span> <a href="quantal1.html#K9" title="QUANTAL1:func.9">delta</a> <font color="Maroon" title="c2">a</font> = 
 <a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> <font color="Olive" title="b1">b</font></span>)</span> 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> ,<font color="Maroon" title="c1">Q</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p2">(<span class="default"><a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quantal1.html#T25" target="_self" title="QUANTAL1:th.25">Th25</a></i>
<br/>.= 
 <a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p4">(<span class="default"><a href="quantal1.html#K5" title="QUANTAL1:func.5">Bottom</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> ,<font color="Maroon" title="c1">Q</font></span>)</span>
<b>by </b><i><a class="txt" href="quantal1.html#E2:68"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="quantal1.html#D6" target="_self" title="QUANTAL1:def.6">Def6</a></i>
<br/>.= 
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="quantal1.html#K9" title="QUANTAL1:func.9">delta</a> <font color="Maroon" title="c2">a</font></span>)</span> 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> ,<font color="Maroon" title="c1">Q</font>
<b>by </b><i><a class="txt" href="quantal1.html#E3:68"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="quantal1.html#E5:68"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="quantal1.html#T24" target="_self" title="QUANTAL1:th.24">Th24</a></i>
;<br/>


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