<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">j</font> being  <a href="quantal1.html#NM1" title="QUANTAL1:NM.1">UnOp</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">j</font> is  <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a>  holds <br/>( <font color="Olive" title="b1">j</font> is  <a href="quantal1.html#V15" title="QUANTAL1:attr.15">\/-distributive</a>  iff for <font color="Olive" title="b2">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Olive" title="b1">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Olive" title="b2">X</font></span>)</span> <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">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">a</font></span>)</span> where <font color="Olive" title="b3">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">j</font> be   <a href="quantal1.html#NM1" title="QUANTAL1:NM.1">UnOp</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">j</font> is  <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a>  implies ( <font color="Maroon" title="c2">j</font> is  <a href="quantal1.html#V15" title="QUANTAL1:attr.15">\/-distributive</a>  iff for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Olive" title="b1">X</font></span>)</span> <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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:36"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c2">j</font> is  <a href="quantal1.html#V14" title="QUANTAL1:attr.14">monotone</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">j</font> is  <a href="quantal1.html#V15" title="QUANTAL1:attr.15">\/-distributive</a>  iff for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Olive" title="b1">X</font></span>)</span> <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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> )</span><br/>

<span class="kw">thus </span><a NAME="E2:36"/>
( <font color="Maroon" title="c2">j</font> is  <a href="quantal1.html#V15" title="QUANTAL1:attr.15">\/-distributive</a>  implies for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Olive" title="b1">X</font></span>)</span> <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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Olive" title="b1">X</font></span>)</span> <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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> </span>}</span> ,<font color="Maroon" title="c1">L</font> ) implies <font color="Maroon" title="c2">j</font> is  <a href="quantal1.html#V15" title="QUANTAL1:attr.15">\/-distributive</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="quantal1.html#D13" target="_self" title="QUANTAL1:def.13">QUANTAL1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Olive" title="b1">X</font></span>)</span> <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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> </span>}</span> ,<font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span> <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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">a</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">L</font></span><br/>

<a NAME="E2:36_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a> <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span>  or <font color="Maroon" title="c4">x</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:36_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E3:36_1_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">and </span><br/><a NAME="E4:36_1_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>
<a NAME="E5:36_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:36_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E3:36_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="quantal1.html#D12" target="_self" title="QUANTAL1:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A5</font></span>: 
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">a</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">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></span>;<br/>
<a NAME="E4:36_1"/>
<font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">a</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">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:36_1"/>
<font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span> <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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">a</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">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:36"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Olive" title="b1">X</font></span>)</span> <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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span> where <font color="Olive" title="b2">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> </span>}</span> ,<font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">j</font> is  <a href="quantal1.html#V15" title="QUANTAL1:attr.15">\/-distributive</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="quantal1.html#D13" target="_self" title="QUANTAL1:def.13">QUANTAL1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="lattices.html#R1" title="LATTICES: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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">a</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">L</font></span><br/>

<a NAME="E4:36"/>
<font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:36"/>
<font color="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="lattice3.html#NK17" title="LATTICE3:NK.17">"\/"</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="lattices.html#R1" title="LATTICES: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="Maroon" title="c2">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">a</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">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E15">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
