<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">P1</font> be    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">P2</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:8"/><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">P1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P1</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P2</font> #)
 ;<br/>

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font>;<br/><b>let </b><font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P2</font>;<br/>





<b>assume </b><a NAME="E2:8"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 ;<br/>

<b>thus </b><a NAME="E3:8"/>
( <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a1</font> implies <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">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">E30</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a1</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>let </b><font color="Maroon">b2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E3:8_1"/>
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E4:8_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:8_1"/>
<font color="Maroon">b2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a2</font>
 <b>by </b><i>, , </i>;<br/>


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

<b>assume </b><a NAME="E4:8"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">a1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>let </b><font color="Maroon">b2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">P1</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E6:8"/>
<font color="Maroon">b2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E7:8"/><b>then </b>
<font color="Maroon">a1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:8"/>
<font color="Maroon">a2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b2</font>
 <b>by </b><i>, , </i>;<br/>


</div>
