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

<b>set </b><font color="Maroon" title="c3">N</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> ;<br/>
<a NAME="E1:30_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T7" title="YELLOW_9:th.7">YELLOW_9:7</a></i>;<br/>
<b>thus </b><a NAME="E2:30_1_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a>  is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c6">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </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/>

<a NAME="E1:30_1_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:30_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1_1"/>
(  not <font color="Maroon" title="c4">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> or  not <font color="Maroon" title="c5">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> or <font color="Maroon" title="c4">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#D1" title="YELLOW_0:def.1">YELLOW_0:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E3:30_1_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a>  is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5" title="YELLOW_6:def.5">YELLOW_6:def 5</a><br/>

<a NAME="E1:30_1_1_2"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:30_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:30_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c5">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">i</font> )
 
;<br/>
<b>hence </b><a NAME="E3:30_1_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> st <br/>( <font color="Maroon" title="c4">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c5">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>


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

<b>thus </b><a NAME="E4:30_1_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a>  is <a href="waybel_0.html#V8" title="WAYBEL_0:attr.8">monotone</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D9" title="WAYBEL_0:def.9">WAYBEL_0:def 9</a>,<a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a><br/>

<a NAME="E1:30_1_1_3"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:30_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1_3"/>
(  not <font color="Maroon" title="c4">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> or <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span>,<font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span>,<font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">j</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#D1" title="YELLOW_0:def.1">YELLOW_0:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel21.html#D2" target="_self" title="WAYBEL21:def.2">WAYBEL21:def 2</a><br/>

<a NAME="E5:30_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E1:30_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:30_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">j</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#D1" title="YELLOW_0:def.1">YELLOW_0:def 1</a></i>;<br/>


</div>
