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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="quantal1.html#NM2">Quantale</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:49"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D17" target="_self">QUANTAL1:def 17</a><br/>

<a NAME="E2:49"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <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">c<sub>1</sub></font>] means <font color="Maroon">a<sub>1</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>] means <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">c<sub>3</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E3:49"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D4">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">c<sub>5</sub></font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>5</sub></font>] )
 <b>by </b><i><a class="txt" href="quantal1.html#E2:49"><i><font color="Green">E22</font></i></a>, <a class="ref" href="quantal1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:49"/><i><font color="Green">E24</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>(<a class="txt" href="quantal1.html#E3:49"><i><font color="Green">E23</font></i></a>);<br/></i>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="quantal1.html#E4:49"><i><font color="Green">E24</font></i></a></i>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>] means <font color="Maroon">c<sub>3</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>] means <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E6:49"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] iff <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:49_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E2:49_3"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>5</sub></font>] iff <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>5</sub></font>] )
 <b>by </b><i><a class="txt" href="quantal1.html#E2:49"><i><font color="Green">E22</font></i></a>, <a class="ref" href="quantal1.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:49"/><i><font color="Green">E26</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>(<a class="txt" href="quantal1.html#E6:49"><i><font color="Green">E25</font></i></a>);<br/></i>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>3</sub></font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="quantal1.html#K1" target="_self">-r&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="quantal1.html#K2" target="_self">-l&gt;</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="quantal1.html#E7:49"><i><font color="Green">E26</font></i></a></i>
;<br/>


</div>
