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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</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">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>set </b><font color="Maroon" title="c4">aN</font> = <font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font>;<br/>
<a NAME="E1:17"/>
<font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</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="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a><br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c10">z1</font> = <font color="Maroon" title="c7">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="ref" href="waybel_2.html#T22" title="WAYBEL_2:th.22">WAYBEL_2:22</a></i>;<br/>
<a NAME="E3:17_1"/><i><font color="Green" title="E15">A2</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="c3">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">N</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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel_2.html#D3" title="WAYBEL_2:def.3">WAYBEL_2:def 3</a></i>;<br/>
<a NAME="E4:17_1"/><b>then </b>
( <font color="Maroon" title="c8">x1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">y1</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">z1</font> )
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:17_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<a NAME="E5:17_1"/><b>then </b>
<font color="Maroon" title="c8">x1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">z1</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:17_1"/>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E3:17_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:17"/>
<font color="Maroon" title="c2">a</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c3">N</font> is   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>
 ;<br/>


</div>
