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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<a NAME="E1:41"/><i><font color="Green">E24</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> #) <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K3">LattPOSet</a> <span class="p1">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#D15">LATTICE3:def 15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> =  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <span class="p2">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:41"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:41"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D12">LATTICE3:def 12</a></i>;<br/>
<a NAME="E6:41"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#R1" target="_self">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E4:41"><i><font color="Green">E29</font></i></a>, <a class="txt" href="yellow_0.html#E5:41"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<a NAME="E7:41"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:41"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:41"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T31">LATTICE3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> <a href="lattice3.html#K4">%</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>assume </b><a NAME="E2:41_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E3:41_1"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a</font> <a href="lattice3.html#K4">%</a> 
 <b>by </b><i><a class="ref" href="lattice3.html#T30">LATTICE3:30</a></i>;<br/><a NAME="E4:41_1"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:41_1"/><b>then </b>
 <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a'</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E6:41"><i><font color="Green">E31</font></i></a>, </i>;<br/><a NAME="E6:41_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K4">%</a>  &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> &amp;  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:41_1"/>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:41"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E9:41"><i><font color="Green">E32</font></i></a>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<b>hence </b><a NAME="E12:41"/>
 <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">y</font> =  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E14:41"/>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<a NAME="E15:41"/><b>then </b>
( <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T28">LATTICE3:28</a></i>;<br/>
<a NAME="E16:41"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <span class="p2">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i/>;<br/><b>assume </b><a NAME="E2:41_2"/>
<font color="Maroon">a</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E3:41_2"/><b>then </b>
<font color="Maroon">a'</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_2"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">a'</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T29">LATTICE3:29</a></i>;<br/><a NAME="E5:41_2"/><b>then </b>
(  <a href="lattice3.html#K5">%</a> <font color="Maroon">a'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span> &amp;  <a href="lattice3.html#K5">%</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a'</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">a'</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/><a NAME="E6:41_2"/><b>then </b>
( <font color="Maroon">a'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a>  &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/><b>hence </b><a NAME="E7:41_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E18:41"/><b>then </b>
 <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E16:41"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E19:41"/>
 <a href="yellow_0.html#K2" target="_self">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K14">latt</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_0.html#E16:41"><i><font color="Green">E33</font></i></a>, , </i>;<br/>


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