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<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">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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="E1:49"/>
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">d</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">d</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D17" target="_self" title="QUANTAL1:def.17">QUANTAL1:def 17</a><br/>

<a NAME="E2:49"/><b>then </b><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">a</font> )
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>] <b>means </b>$1 <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>] <b>means </b>$1 <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E3:49"/><i><font color="Green" title="E23">A2</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">d</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">d</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>;<br/>

<a NAME="E1:49_1"/>
<font color="Maroon" title="c5">d</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E2:49_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">d</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c5">d</font>] )
 <b>by </b><i><a class="txt" href="quantal1.html#E2:49"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="quantal1.html#T12" target="_self" title="QUANTAL1:th.12">Th12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:49"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">d1</font>) where <font color="Olive" title="b1">d1</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">d1</font>] </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="Olive" title="b1">d2</font>) where <font color="Olive" title="b1">d2</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>2</sub></font>[<font color="Olive" title="b1">d2</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3" title="FRAENKEL:sch.3">FRAENKEL:sch 3</a>(<a class="txt" href="quantal1.html#E3:49"><i><font color="Green" title="E23">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:49"/>
<font color="Maroon" title="c3">b</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>] <b>means </b><font color="Maroon" title="c3">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> $1 <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> $1 <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E6:49"/><i><font color="Green" title="E24">A3</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> holds <br/> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">d</font>] iff <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">d</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>;<br/>

<a NAME="E1:49_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E2:49_2"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c5">d</font>] iff <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c5">d</font>] )
 <b>by </b><i><a class="txt" href="quantal1.html#E2:49"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="quantal1.html#T11" target="_self" title="QUANTAL1:th.11">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:49"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">d1</font>) where <font color="Olive" title="b1">d1</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>3</sub></font>[<font color="Olive" title="b1">d1</font>] </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="Olive" title="b1">d2</font>) where <font color="Olive" title="b1">d2</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>4</sub></font>[<font color="Olive" title="b1">d2</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3" title="FRAENKEL:sch.3">FRAENKEL:sch 3</a>(<a class="txt" href="quantal1.html#E6:49"><i><font color="Green" title="E24">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:49"/>
<font color="Maroon" title="c3">b</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">a</font>
 ;<br/>


</div>
