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





<b>set </b><font color="Maroon">c<sub>4</sub></font> = <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="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> ;<br/>
<a NAME="E1:71"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> iff ( <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="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">c<sub>2</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"> <font color="Olive">b<sub>2</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>2</sub></font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</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>2</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>thus </b><a NAME="E2:71"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies ( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R3" 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="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:71_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:71_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <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="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">c<sub>2</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"> <font color="Olive">b<sub>2</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>2</sub></font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</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>2</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>thus </b><a NAME="E3:71_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" target="_self">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E1:71_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:71_1_1"/>
<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="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" target="_self">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:71_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</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="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:71_1_1_1"/><b>then </b>
ex <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 <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:71_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E1:71_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattice3.html#D16" target="_self">Def16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:71_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E1:71_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="lattice3.html#D21" target="_self">Def21</a></i>;<br/>


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

<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/>

<b>assume </b><a NAME="E4:71_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E5:71_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</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="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E6:71_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E2:71_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="lattice3.html#D17" target="_self">Def17</a></i>;<br/>


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

<b>assume </b><a NAME="E3:71"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R3" 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="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 ;<br/>

<a NAME="E4:71"/><i><font color="Green">E43</font></i>: 
<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="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" target="_self">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:71_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</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="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:71_2"/><b>then </b>
ex <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 <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:71_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E3:71"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:71"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</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="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice3.html#E3:71"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:71"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="lattice3.html#E1:71"><i><font color="Green">E41</font></i></a>, <a class="txt" href="lattice3.html#E4:71"><i><font color="Green">E43</font></i></a>, <a class="ref" href="lattice3.html#D17" target="_self">Def17</a></i>;<br/>


</div>
