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<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#K3" title="WAYBEL_9:func.3">+id</a> ;<br/>
<b>thus </b><a NAME="E1:31_1_1"/>
<font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K3" title="WAYBEL_9:func.3">+id</a>  is <a href="waybel_0.html#V8" title="WAYBEL_0:attr.8">monotone</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#K3" title="WAYBEL_9:func.3">+id</a>  is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1_1"/><i><font color="Green" title="E16">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#K3" title="WAYBEL_9:func.3">+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#D5" target="_self" title="WAYBEL_9:def.5">Def5</a></i>;<br/>
<a NAME="E2:31_1_1_1"/>
 <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#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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 <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> #)
 
<b>by </b><i><a class="ref" href="waybel_9.html#D5" target="_self" title="WAYBEL_9:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:31_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#K3" title="WAYBEL_9:func.3">+id</a> </span>)</span>,<font color="Maroon" title="c1">L</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="txt" href="#E1:31_1_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="waybel_9.html#T1" target="_self" title="WAYBEL_9:th.1">Th1</a></i>; <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="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="E2:31_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#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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#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#K3" title="WAYBEL_9:func.3">+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="31_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#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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#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#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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#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#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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#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#K3" title="WAYBEL_9:func.3">+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="E1:31_1_1_2"/><i><font color="Green" title="E16">A2</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#K3" title="WAYBEL_9:func.3">+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#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#K3" title="WAYBEL_9:func.3">+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/>

<a NAME="E2:31_1_1_2"/><i><font color="Green" title="E17">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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K3" title="WAYBEL_9:func.3">+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#K3" title="WAYBEL_9:func.3">+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 <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> #)
 
<b>by </b><i><a class="ref" href="waybel_9.html#D5" target="_self" title="WAYBEL_9:def.5">Def5</a></i>;<br/>
<a NAME="E3:31_1_1_2"/>
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#K3" title="WAYBEL_9:func.3">+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#D5" target="_self" title="WAYBEL_9:def.5">Def5</a></i>;<br/>
<a NAME="E4:31_1_1_2"/><b>then </b>
( 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#K3" title="WAYBEL_9:func.3">+id</a> </span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</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="c1">L</font> <a href="waybel_9.html#K3" title="WAYBEL_9:func.3">+id</a> </span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> )
 
<b>by </b><i><a class="txt" href="#E2:31_1_1_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="tmap_1.html#T91" title="TMAP_1:th.91">TMAP_1:91</a></i>;<br/>
<b>hence </b><a NAME="E5:31_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K3" title="WAYBEL_9:func.3">+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#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#K3" title="WAYBEL_9:func.3">+id</a> </span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="#E1:31_1_1_2"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="#E2:31_1_1_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</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="E3:31_1_1"/>
<font color="Maroon" title="c1">L</font> <a href="waybel_9.html#K3" title="WAYBEL_9:func.3">+id</a>  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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
