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<b>let </b><font color="Maroon">L</font> be  <a href="orders_2.html#V3">transitive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V1">with_suprema</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">s</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">p</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">s</font> holds <br/><font color="Maroon">q</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">s</font>
 <b>and </b><br/><a NAME="E3:33"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">u</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E4:33"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="waybel16.html#T1" target="_self">Th1</a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E5:33"/>
<font color="Maroon">q</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E6:33"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">q</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E7:33"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">q</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:33_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">v</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:33_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">v</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font>
 ;<br/><a NAME="E3:33_1"/>
<font color="Maroon">v</font> <a href="orders_2.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="waybel16.html#E2:33"><i><font color="Green">E26</font></i></a>, , <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><a NAME="E4:33_1"/><b>then </b>
<font color="Maroon">v</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:33_1"/>
<font color="Maroon">q</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:33"/>
<font color="Maroon">p</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="waybel16.html#T3" target="_self">Th3</a>, <a class="txt" href="waybel16.html#E1:33"><i><font color="Green">E25</font></i></a>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>


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