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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">N</font> be  <a href="waybel_0.html#V10">eventually-directed</a> <a href="waybel_0.html#NM2">net</a> of  <a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>





<b>set </b><font color="Maroon">M</font> =  <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">L</font> = <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E1:49"/>
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>  ex <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> st <br/>for <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>  st <font color="Olive">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/><span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>;<br/>

<a NAME="E1:49_1"/><i><font color="Green">E42</font></i>: 
 <a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> is    <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="waybel24.html#D3">WAYBEL24:def 3</a></i>;<br/>
<a NAME="E2:49_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E3:49_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> #)
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E6:49_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">c</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T11">WAYBEL_0:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">j</font>
;<br/>

<b>let </b><font color="Maroon">k</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Na</font> = <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a</font>, <font color="Maroon">Nc</font> = <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">Na</font>, <font color="Maroon">C</font> = <font color="Maroon">Nc</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<b>assume </b><a NAME="E11:49_1"/>
<font color="Maroon">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E12:49_1"/><b>then </b>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<a NAME="E13:49_1"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:49_1"/><b>then </b>
<font color="Maroon">Na</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Nc</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/>
<a NAME="E15:49_1"/><b>then </b>
<font color="Maroon">A</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E16:49_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">A</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">C</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>
<a NAME="E17:49_1"/><i><font color="Green">E54</font></i>: 
( the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E18:49_1"/>
<span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:49"/>
 <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> is <a href="waybel_0.html#V10">eventually-directed</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#T11">WAYBEL_0:11</a></i>;<br/>


</div>
