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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V4" target="_self">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/>

<b>thus </b><a NAME="E1:70"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="lattice3.html#V5" target="_self">\/-distributive</a> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> 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>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Olive">b<sub>5</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>5</sub></font> ) &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>5</sub></font></span>)</span> 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>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Olive">b<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>6</sub></font></span>)</span> 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>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Olive">b<sub>5</sub></font> holds <br/><font color="Olive">b<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>5</sub></font> ) holds <br/><font color="Olive">b<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>4</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D19" target="_self">LATTICE3:def 19</a><br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> 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="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ;<br/>
<a NAME="E2:70_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4" target="_self">is_less_than</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; ( 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> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> 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="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> 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="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font> &amp; ( 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> st <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> 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>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> 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>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="lattice3.html#D21" target="_self">Def21</a></i>;<br/>
<b>hence </b><a NAME="E3:70_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> 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="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E1:70_1"><i><font color="Green">E40</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:70"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>3</sub></font></span>)</span> 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>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D19" target="_self">LATTICE3:def 19</a><br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <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/>







<b>assume </b><a NAME="E3:70"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( 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> st <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> 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="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( 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> st <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> 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>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<a NAME="E4:70"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> 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="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> ,<font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#D21" target="_self">Def21</a></i>;<br/>
<b>hence </b><a NAME="E5:70"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E2:70"><i><font color="Green">E40</font></i></a></i>;<br/>


</div>
