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<b>let </b><font color="Maroon">Q</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattices.html#V10">Lattice-like</a> <a href="lattice3.html#V4">complete</a> <a href="quantal1.html#V7" target="_self">right-distributive</a> <a href="quantal1.html#V8" target="_self">left-distributive</a>  <a href="quantal1.html#L1" target="_self">QuantaleStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font>;<br/>







<b>set </b><font color="Maroon">X1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">d</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">X2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span> ;<br/>
<a NAME="E1:22"/>
<font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#NK17" target="_self">"\/"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="lattice3.html#T44">LATTICE3:44</a></i>;<br/>
<a NAME="E2:22"/><b>then </b><i><font color="Green">E26</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font> <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">d</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span> ,<font color="Maroon">Q</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="lattices.html#K3">"\/"</a> <font color="Maroon">b</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="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span> ,<font color="Maroon">Q</font> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:22_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E2:22_1"/>
( ( <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">c</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font> ) implies <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">d</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span>  )
 ;<br/><b>assume </b><a NAME="E3:22_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">d</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E5:22_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> )
 ;<br/><b>thus </b><a NAME="E6:22_1"/>
( <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">c</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:22"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">d</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><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">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:22"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">c</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">c</font></span>)</span> <a href="lattices.html#K3">"\/"</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>
 <b>by </b><i>, <a class="ref" href="lattice3.html#T44">LATTICE3:44</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:22_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E2:22_2"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font> ) implies <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="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span>  )
 ;<br/><b>assume </b><a NAME="E3:22_2"/>
<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="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font><b> such that </b><br/><a NAME="E5:22_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> )
 ;<br/><b>thus </b><a NAME="E6:22_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:22"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Olive">d</font></span>)</span> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Q</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:22"/>
<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="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</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#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="monoid_0.html#NK3" target="_self">[*]</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="lattice3.html#T44">LATTICE3:44</a></i>;<br/>


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