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<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="quantal1.html#NM1">UnOp</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">j</font> is <a href="quantal1.html#V14" target="_self">monotone</a>
 ;<br/>

<b>thus </b><a NAME="E2:36"/>
( <font color="Maroon">j</font> is <a href="quantal1.html#V15" target="_self">\/-distributive</a> implies for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> holds  <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">X</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">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> </span>}</span> ,<font color="Maroon">L</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> holds  <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">X</font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> </span>}</span> ,<font color="Maroon">L</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D13" target="_self">QUANTAL1:def 13</a><br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<a NAME="E2:36_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:36_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span>  <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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="E1:36_1_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="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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">L</font><b> such that </b><br/><a NAME="E3:36_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>
<a NAME="E4:36_1_1"/>
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T38">LATTICE3:38</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_1"/>
<font color="Maroon">x</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:36_1"/><b>then </b>
 <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1"/>
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</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">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:36"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> holds  <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Olive">X</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">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> </span>}</span> ,<font color="Maroon">L</font>
 ;<br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D13" target="_self">QUANTAL1:def 13</a><br/>

<a NAME="E4:36"/>
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:36"/>
<font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" target="_self">"\/"</a> <font color="Maroon">X</font></span>)</span> <a href="lattices.html#R1">[=</a>  <a href="lattice3.html#K15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div>
