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

<b>set </b><font color="Maroon" title="c2">N</font> = <font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> ;<br/>
<b>thus </b><a NAME="E1:39_1_1"/>
<font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a>  is <a href="waybel_0.html#V9" title="WAYBEL_0:attr.9">antitone</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a>  is <a href="waybel_0.html#V11" title="WAYBEL_0:attr.11">eventually-filtered</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_1_1"/><i><font color="Green" title="E19">A1</font></i>: 
 <a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span>,<font color="Maroon" title="c1">L</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#D6" target="_self" title="WAYBEL_9:def.6">Def6</a></i>;<br/>
<a NAME="E2:39_1_1_1"/><i><font color="Green" title="E20">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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </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="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel_9.html#T11" target="_self" title="WAYBEL_9:th.11">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">f</font> =  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>,<font color="Maroon" title="c1">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span> ;<br/>
<a NAME="E5:39_1_1_1"/><i><font color="Green" title="E21">A3</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="c1">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</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 <font color="Maroon" title="c1">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> #)
 
;<br/>
<a NAME="E6:39_1_1_1"/>
<font color="Maroon" title="c4">g</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T40" title="YELLOW_7:th.40">YELLOW_7:40</a></i>;<br/>
<a NAME="E7:39_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T41" title="YELLOW_7:th.41">YELLOW_7:41</a></i>;<br/>
<b>hence </b><a NAME="E8:39_1_1_1"/>
 <a href="waybel_0.html#K1" title="WAYBEL_0:func.1">netmap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span>,<font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a>
 <b>by </b><i><a class="txt" href="waybel_9.html#E1:39_1_1_1"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="waybel_9.html#E2:39_1_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="waybel_9.html#E5:39_1_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="waybel_9.html#T2" target="_self" title="WAYBEL_9:th.2">Th2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D10" title="WAYBEL_0:def.10">WAYBEL_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<a NAME="E2:39_1_1"/>
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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </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#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </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="39_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> st <br/>for <font color="Olive" title="b2">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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span>  st <font color="Olive" title="b1">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">k</font></span><br/>

<b>take </b><font color="Maroon" title="c4">j</font> = <font color="Maroon" title="c3">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span>  st <font color="Maroon" title="c4">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font></span><br/>

<b>let </b><font color="Maroon" title="c5">k</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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">k</font> )</span><br/>

<b>assume </b><a NAME="E1:39_1_1_2"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">k</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">i1</font> = <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c7">k1</font> = <font color="Maroon" title="c5">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel_9.html#D6" target="_self" title="WAYBEL_9:def.6">Def6</a></i>;<br/>
<a NAME="E3:39_1_1_2"/><i><font color="Green" title="E20">A5</font></i>: 
( <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c6">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i1</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c7">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k1</font> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T91" title="TMAP_1:th.91">TMAP_1:91</a></i>;<br/>
<a NAME="E4:39_1_1_2"/><i><font color="Green" title="E21">A6</font></i>: 
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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#D6" target="_self" title="WAYBEL_9:def.6">Def6</a></i>;<br/>
<a NAME="E5:39_1_1_2"/><i><font color="Green" title="E22">A7</font></i>: 
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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> 
 
<b>by </b><i><a class="ref" href="waybel_9.html#D6" target="_self" title="WAYBEL_9:def.6">Def6</a></i>;<br/>
<a NAME="E6:39_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c5">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:39_1_1_2"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E7:39_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c3">i</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E8:39_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E3:39_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="waybel_9.html#E4:39_1_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="waybel_9.html#E5:39_1_1_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:39_1_1"/>
<font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K4" title="WAYBEL_9:func.4">opp+id</a>  is <a href="waybel_0.html#V11" title="WAYBEL_0:attr.11">eventually-filtered</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#T12" title="WAYBEL_0:th.12">WAYBEL_0:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
