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<div><i><font color="Green" title="E8">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">L</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</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>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:13_1"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font>
 ;<br/><b>thus </b><a NAME="E2:13_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E2:13_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">a</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 
;<br/>
<a NAME="E3:13_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="txt" href="yellow_7.html#E1:13_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="yellow_7.html#E1:13_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:13_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> 
 <b>by </b><i><a class="txt" href="yellow_7.html#E2:13_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="lattice3.html#T9" title="LATTICE3:th.9">LATTICE3:9</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green" title="E9">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">L</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</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>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:13_2"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font>
 ;<br/><b>thus </b><a NAME="E2:13_2"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:13_2_1"/><i><font color="Green" title="E11">A7</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E2:13_2_1"/><i><font color="Green" title="E12">A8</font></i>: 
( <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">a</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> )
 
;<br/>
<a NAME="E3:13_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="txt" href="yellow_7.html#E1:13_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="yellow_7.html#E1:13_2_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:13_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="yellow_7.html#E2:13_2_1"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="lattice3.html#T9" title="LATTICE3:th.9">LATTICE3:9</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c1">L</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</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>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<a NAME="E3:13"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c2">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T2" title="YELLOW_0:th.2">YELLOW_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:13"/>
( <font color="Maroon" title="c2">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> iff <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:13"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="yellow_7.html#E2:13"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/>

<a NAME="E5:13"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T2" title="YELLOW_0:th.2">YELLOW_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:13"/>
( <font color="Maroon" title="c2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">X</font> iff <font color="Maroon" title="c2">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="txt" href="yellow_7.html#E1:13"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="yellow_7.html#E2:13"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/>


</div>
