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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>thus </b><a NAME="E1:7"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V3">transitive</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is <a href="orders_2.html#V3">transitive</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V3">transitive</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a><br/>

<b>assume </b><a NAME="E2:7_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E3:7_1"/><b>then </b>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_7.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:7_1"/><b>then </b>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_7.html#E1:7_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:7_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_7.html#T1" target="_self">Th1</a></i>;<br/>


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

<b>assume </b><a NAME="E2:7"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is <a href="orders_2.html#V3">transitive</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a><br/>

<b>assume </b><a NAME="E3:7"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E4:7"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K8">~</a>  )
 
<b>by </b><i><a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>
<a NAME="E5:7"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattice3.html#K8">~</a> 
 
<b>by </b><i><a class="txt" href="yellow_7.html#E2:7"><i><font color="Green">E6</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:7"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>


</div>
