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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">q</font>
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green" title="E21">A2</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">p</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Maroon" title="c3">q</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">s</font>
 <b>and </b><br/><a NAME="E3:33"/><i><font color="Green" title="E22">A3</font></i>: 
not <font color="Maroon" title="c4">u</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font>
 ;<br/>

<a NAME="E4:33"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">q</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E1:33"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E5:33"/>
<font color="Maroon" title="c3">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">u</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">q</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/>
<a NAME="E6:33"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c3">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">u</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="waybel16.html#E4:33"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<a NAME="E7:33"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c3">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">u</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">u</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">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" title="c5">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:33_1"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font>
 <b>and </b><br/><a NAME="E2:33_1"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">u</font>
 ;<br/><a NAME="E3:33_1"/>
<font color="Maroon" title="c5">v</font> <a href="orders_2.html#NR4" title="ORDERS_2:NR.4">&gt;</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E3:33"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="waybel16.html#E1:33_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="waybel16.html#E2:33_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><a NAME="E4:33_1"/><b>then </b>
<font color="Maroon" title="c5">v</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">q</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E2:33"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E5:33_1"/>
<font color="Maroon" title="c3">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">u</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">v</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E2:33_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:33"/>
<font color="Maroon" title="c2">p</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">u</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E6:33"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="waybel16.html#E7:33"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/>


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