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<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>;<br/>





<b>thus </b><a NAME="E1:61"/>
( <font color="Maroon" title="c3">p</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c3">p</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c1">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green" title="E35">A1</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c3">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">q</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" target="_self" title="LATTICE3:def.16">LATTICE3:def 16</a><br/>

<b>let </b><font color="Maroon" title="c4">p'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c2">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" target="_self" title="LATTICE3:def.8">LATTICE3:def 8</a><br/>

<a NAME="E2:61_1"/><i><font color="Green" title="E36">A2</font></i>: 
( <font color="Maroon" title="c4">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c4">p'</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c4">p'</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c4">p'</font> )
 
;<br/>
<b>assume </b><a NAME="E3:61_1"/>
<font color="Maroon" title="c4">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E4:61_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c4">p'</font>
 
<b>by </b><i><a class="txt" href="lattice3.html#E1:61_1"><i><font color="Green" title="E35">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:61_1"/>
<font color="Maroon" title="c3">p</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">p'</font>
 <b>by </b><i><a class="txt" href="lattice3.html#E2:61_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></i>;<br/>


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

<b>assume </b><a NAME="E2:61"/><i><font color="Green" title="E35">A3</font></i>: 
for <font color="Olive" title="b1">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c2">L</font></span>)</span>  st <font color="Olive" title="b1">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c3">p</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">q'</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" target="_self" title="LATTICE3:def.8">LATTICE3:def 8</a><br/>

<b>let </b><font color="Maroon" title="c4">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" target="_self" title="LATTICE3:def.16">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E3:61"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E4:61"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">q</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
<b>by </b><i><a class="txt" href="lattice3.html#E2:61"><i><font color="Green" title="E35">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:61"/>
<font color="Maroon" title="c3">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></i>;<br/>


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