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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V1">with_suprema</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <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/>



<a NAME="E1:30"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E2:30"/><b>then </b><i><font color="Green">E20</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a>  )
 
<b>by </b><i><a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>
<a NAME="E3:30"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="lattice3.html#K9">~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="lattice3.html#K9">~</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a>  )
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>;<br/><b>assume </b><a NAME="E1:30_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a>  )
 ;<br/><a NAME="E2:30_1"/><b>then </b>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="yellow_7.html#E3:30"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow_7.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:30_1"/><b>then </b>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><b>hence </b><a NAME="E4:30_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:30"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_7.html#E2:30"><i><font color="Green">E20</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E6:30"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a> </span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a> </span>)</span>
 ;<br/>


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