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<b>let </b><font color="Maroon" title="c1">P1</font>, <font color="Maroon" title="c2">P2</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E3">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">P1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">P1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">P2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">P2</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>let </b><font color="Maroon" title="c4">a1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P1</font>;<br/><b>let </b><font color="Maroon" title="c5">a2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">P2</font>;<br/>





<b>assume </b><a NAME="E2:8"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a2</font>
 ;<br/>

<b>thus </b><a NAME="E3:8"/>
( <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c4">a1</font> implies <font color="Maroon" title="c3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">a2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/><i><font color="Green" title="E5">A3</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P1</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a1</font>
 ; <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>let </b><font color="Maroon" title="c6">b2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">P2</font>; <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>reconsider </b><font color="Maroon" title="c7">b1</font> = <font color="Maroon" title="c6">b2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P1</font> <b>by </b><i><a class="txt" href="yellow_0.html#E1:8"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:8_1"/>
<font color="Maroon" title="c6">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E4:8_1"/><b>then </b>
<font color="Maroon" title="c7">b1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">a1</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E1:8_1"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:8_1"/>
<font color="Maroon" title="c6">b2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">a2</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:8"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="yellow_0.html#E2:8"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="yellow_0.html#T1" target="_self" title="YELLOW_0:th.1">Th1</a></i>;<br/>


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

<b>assume </b><a NAME="E4:8"/><i><font color="Green" title="E5">A4</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P1</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon" title="c4">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">b</font>
 ; <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>let </b><font color="Maroon" title="c6">b2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">P2</font>; <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>reconsider </b><font color="Maroon" title="c7">b1</font> = <font color="Maroon" title="c6">b2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P1</font> <b>by </b><i><a class="txt" href="yellow_0.html#E1:8"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E6:8"/>
<font color="Maroon" title="c6">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/>

<a NAME="E7:8"/><b>then </b>
<font color="Maroon" title="c4">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">b1</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E4:8"><i><font color="Green" title="E5">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:8"/>
<font color="Maroon" title="c5">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">b2</font>
 <b>by </b><i><a class="txt" href="yellow_0.html#E1:8"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="yellow_0.html#E2:8"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="yellow_0.html#T1" target="_self" title="YELLOW_0:th.1">Th1</a></i>;<br/>


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