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<b>let </b><font color="Maroon">L</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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<a NAME="E1:55"/>
( <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="txt" href="yellow_0.html#E2:55"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E2:55"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon">c</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E2:55_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:55_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E2:55"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:55"/><b>then </b>
 <a href="yellow_0.html#R1" target="_self">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:55"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/>


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