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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a> <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM1" title="WAYBEL_0:NM.1">prenet</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">x</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>assume </b><a NAME="E1:40"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c2">N</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 ;<br/>

<a NAME="E2:40"/><i><font color="Green" title="E27">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 <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">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="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span> #) <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">N</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">N</font> #)
 
<b>by </b><i><a class="ref" href="waybel_2.html#D3" target="_self" title="WAYBEL_2:def.3">Def3</a></i>;<br/>
<a NAME="E3:40"/>
for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span>  ex <font color="Olive" title="b2">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span> st <br/>for <font color="Olive" title="b3">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span>  st <font color="Olive" title="b2">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b3">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i1</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="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">i</font> = <font color="Maroon" title="c4">i1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="ref" href="waybel_2.html#T22" target="_self" title="WAYBEL_2:th.22">Th22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green" title="E28">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Maroon" title="c6">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E1:40"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="waybel_0.html#T11" title="WAYBEL_0:th.11">WAYBEL_0:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">j1</font> = <font color="Maroon" title="c6">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_2.html#T22" target="_self" title="WAYBEL_2:th.22">Th22</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">j1</font>
;<br/>

<b>let </b><font color="Maroon" title="c8">k1</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="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c8">k1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="ref" href="waybel_2.html#T22" target="_self" title="WAYBEL_2:th.22">Th22</a></i>;<br/>
<b>assume </b><a NAME="E6:40_1"/>
<font color="Maroon" title="c7">j1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">k1</font>
 ;<br/>

<a NAME="E7:40_1"/><b>then </b>
<font color="Maroon" title="c6">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E2:40"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<a NAME="E8:40_1"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E3:40_1"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<a NAME="E9:40_1"/><i><font color="Green" title="E30">A5</font></i>: 
 ex <font color="Olive" title="b1">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i1</font> &amp; the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">yi</font> )
 
<b>by </b><i><a class="ref" href="waybel_2.html#D3" target="_self" title="WAYBEL_2:def.3">Def3</a></i>;<br/>
<a NAME="E10:40_1"/>
 ex <font color="Olive" title="b1">yk</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">yk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k1</font> &amp; the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">yk</font> )
 
<b>by </b><i><a class="ref" href="waybel_2.html#D3" target="_self" title="WAYBEL_2:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E11:40_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c4">i1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">k1</font>
 <b>by </b><i><a class="txt" href="waybel_2.html#E8:40_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="waybel_2.html#E9:40_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="waybel_1.html#T2" title="WAYBEL_1:th.2">WAYBEL_1:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:40"/>
<font color="Maroon" title="c3">x</font> <a href="waybel_2.html#K3" title="WAYBEL_2:func.3">"/\"</a> <font color="Maroon" title="c2">N</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#T11" title="WAYBEL_0:th.11">WAYBEL_0:11</a></i>;<br/>


</div>
