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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">Y</font> be  <a href="waybel25.html#V1" title="WAYBEL25:attr.1">monotone-convergence</a> <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of  <a href="waybel24.html#K3" title="WAYBEL24:func.3">ContMaps</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:79"/><i><font color="Green" title="E44">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds   <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 <b>and </b><br/><a NAME="E2:79"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">n</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>;<br/>
<b>set </b><font color="Maroon" title="c7">m</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">L</font> = <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E3:79"/><i><font color="Green" title="E46">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:79"/><b>then </b><i><font color="Green" title="E47">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E34"><i><font color="Green" title="E28">Lm6</font></i></a></i>;<br/>
<a NAME="E5:79"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds   <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/>

<a NAME="E1:79_1"/>
 <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c9">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:79"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:79_1"/>
 <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c9">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel25.html#T31" target="_self" title="WAYBEL25:th.31">Th31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:79"/><b>then </b><i><font color="Green" title="E48">A5</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="waybel25.html#T26" target="_self" title="WAYBEL25:th.26">Th26</a></i>;<br/>
<a NAME="E7:79"/><i><font color="Green" title="E49">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c5">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:79_2"/>
<font color="Maroon" title="c9">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:79_2"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:79_2"/><i><font color="Green" title="E51">A8</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">i</font> = <font color="Maroon" title="c10">i</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="waybel25.html#E3:79"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="waybel25.html#E3:79_2"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="waybel25.html#E34"><i><font color="Green" title="E28">Lm6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">g</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c11">i</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E33"><i><font color="Green" title="E27">Lm5</font></i></a></i>;<br/>
<a NAME="E7:79_2"/>
<font color="Maroon" title="c12">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E3:79"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:79_2"/><b>then </b>
<font color="Maroon" title="c12">g</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E2:79"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="waybel25.html#E6:79"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="waybel25.html#T40" target="_self" title="WAYBEL25:th.40">Th40</a></i>;<br/>
<a NAME="E9:79_2"/><b>then </b>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">b</font> )
 
<b>by </b><i><a class="ref" href="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:79_2"/>
<font color="Maroon" title="c9">w</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E4:79_2"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="waybel25.html#T24" target="_self" title="WAYBEL25:th.24">Th24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:79"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:79_3"/><i><font color="Green" title="E50">A9</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>  st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">e</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> implies  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Olive" title="b1">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> ) ) );<br/>
<a NAME="E2:79_3"/><i><font color="Green" title="E51">A10</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">e</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_3_1_1"/>
( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c10">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_3_1_1_1"><b>suppose </b></a><a NAME="E1:79_3_1_1_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:79_3_1_1_1"/>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">e</font>,<font color="Olive" title="b1">u</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_3_1_1_2"><b>suppose </b></a><a NAME="E1:79_3_1_1_2"/><i><font color="Green" title="E52">A11</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c11">e</font> = <font color="Maroon" title="c10">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><b>take </b>
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">e</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
;<br/><b>thus </b><a NAME="E3:79_3_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">e</font>, <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c11">e</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>]
 <b>by </b><i><a class="txt" href="waybel25.html#E1:79_3_1_1_2"><i><font color="Green" title="E52">A11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">t</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span><b> such that </b><br/><a NAME="E4:79_3"/><i><font color="Green" title="E52">A12</font></i>: 
for <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">u</font>,<font color="Maroon" title="c10">t</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">u</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="waybel25.html#E2:79_3"><i><font color="Green" title="E51">A10</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c11">t</font> = <font color="Maroon" title="c10">t</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">tt</font> = <font color="Maroon" title="c11">t</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T19" title="WAYBEL24:th.19">WAYBEL24:19</a></i>;<br/>
<a NAME="E7:79_3"/>
<font color="Maroon" title="c12">tt</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">s</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">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:79_3_2"/>
<font color="Maroon" title="c13">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:79_3_2"/><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font>
 <b>and </b><br/><a NAME="E4:79_3_2"/><i><font color="Green" title="E54">A14</font></i>: 
<font color="Maroon" title="c13">s</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="c3">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">i</font> = <font color="Maroon" title="c14">i</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">N</font> <b>by </b><i><a class="txt" href="waybel25.html#E3:79_3_2"><i><font color="Green" title="E53">A13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">ss</font> = <font color="Maroon" title="c13">s</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T19" title="WAYBEL24:th.19">WAYBEL24:19</a></i>;<br/>
<a NAME="E7:79_3_2"/><i><font color="Green" title="E55">A15</font></i>: 
<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">s'</font> = <font color="Maroon" title="c13">s</font>, <font color="Maroon" title="c18">t'</font> = <font color="Maroon" title="c12">tt</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E9:79_3_2"/>
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c17">s'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">t'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<a NAME="E1:79_3_2_1"/><i><font color="Green" title="E56">A16</font></i>: 
<font color="Maroon" title="c16">ss</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c19">j</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 <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c19">j</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:79_3_2"><i><font color="Green" title="E54">A14</font></i></a>, <a class="ref" href="waybel25.html#T24" target="_self" title="WAYBEL25:th.24">Th24</a></i>;<br/>
<a NAME="E2:79_3_2_1"/><i><font color="Green" title="E57">A17</font></i>: 
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="yellow_1.html#K4" title="YELLOW_1:func.4">.</a> <font color="Maroon" title="c19">j</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_3_2_1_1"/>
( <font color="Maroon" title="c19">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:79_3_2_1_1_1"/>
<font color="Maroon" title="c19">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font>
 ;<br/><div class="add"><a NAME="E2:79_3_2_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c11">t</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Olive" title="b1">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel25.html#E4:79_3"><i><font color="Green" title="E52">A12</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E58">A18</font></i>: <font color="Maroon" title="c18">t'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c19">j</font> = 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_2.html#D1" title="WAYBEL_2:def.1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p2">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c19">j</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span>

;<br/><a NAME="E4:79_3_2_1_1_1"/>
 <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c19">j</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:79"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/><a NAME="E5:79_3_2_1_1_1"/><b>then </b>
 <a href="waybel_9.html#R1" title="WAYBEL_9:pred.1">ex_sup_of</a>  <a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c19">j</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel25.html#T31" target="_self" title="WAYBEL25:th.31">Th31</a></i>;<br/><a NAME="E6:79_3_2_1_1_1"/><b>then </b><i><font color="Green" title="E59">A19</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c19">j</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>, <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="ref" href="waybel_9.html#D3" title="WAYBEL_9:def.3">WAYBEL_9:def 3</a></i>;<br/><a NAME="E7:79_3_2_1_1_1"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c19">j</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel25.html#E3:79_3_2"><i><font color="Green" title="E53">A13</font></i></a>, <a class="txt" href="waybel25.html#E34"><i><font color="Green" title="E28">Lm6</font></i></a></i>;<br/><a NAME="E8:79_3_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">ss</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c19">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c19">j</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:79_3_2_1"><i><font color="Green" title="E56">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:79_3_2_1_1_1"/>
<font color="Maroon" title="c17">s'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c19">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">t'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c19">j</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E2:79_3_2_1"><i><font color="Green" title="E57">A17</font></i></a>, <a class="txt" href="waybel25.html#E3:79_3_2_1_1_1"><i><font color="Green" title="E58">A18</font></i></a>, <a class="txt" href="waybel25.html#E6:79_3_2_1_1_1"><i><font color="Green" title="E59">A19</font></i></a>, <a class="ref" href="yellow_4.html#T1" title="YELLOW_4:th.1">YELLOW_4:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:79_3_2_1_1_2"/><i><font color="Green" title="E58">A20</font></i>: 
<font color="Maroon" title="c19">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 ;<br/><div class="add"><a NAME="E2:79_3_2_1_1_2"/><i><font color="Green" title="E59">A21</font></i>: 
<font color="Maroon" title="c16">ss</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">x</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 <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E4:79_3_2"><i><font color="Green" title="E54">A14</font></i></a>, <a class="ref" href="waybel25.html#T24" target="_self" title="WAYBEL25:th.24">Th24</a></i>;<br/><a NAME="E3:79_3_2_1_1_2"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel25.html#E4:79"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:79_3_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">ss</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:79_3"><i><font color="Green" title="E50">A9</font></i></a>, <a class="txt" href="waybel25.html#E2:79_3_2_1_1_2"><i><font color="Green" title="E59">A21</font></i></a></i>;<br/><b>hence </b><a NAME="E5:79_3_2_1_1_2"/>
<font color="Maroon" title="c17">s'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c19">j</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">t'</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c19">j</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E4:79_3"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="waybel25.html#E2:79_3_2_1"><i><font color="Green" title="E57">A17</font></i></a>, <a class="txt" href="waybel25.html#E1:79_3_2_1_1_2"><i><font color="Green" title="E58">A20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E10:79_3_2"/>
<font color="Maroon" title="c13">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">tt</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E7:79_3_2"><i><font color="Green" title="E55">A15</font></i></a>, <a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:79_3"/><b>then </b><i><font color="Green" title="E53">A22</font></i>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">tt</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E6:79"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>;<br/>
<a NAME="E9:79_3"/><i><font color="Green" title="E54">A23</font></i>: 
<font color="Maroon" title="c12">tt</font> <a href="yellow14.html#K1" title="YELLOW14:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:79_3"><i><font color="Green" title="E52">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">p</font> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">N</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font></span>)</span>, <font color="Maroon" title="c14">q</font> = <font color="Maroon" title="c12">tt</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E11:79_3"/><i><font color="Green" title="E55">A24</font></i>: 
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="yellow_1.html#K4" title="YELLOW_1:func.4">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E12:79_3"/>
<font color="Maroon" title="c13">p</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">q</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E8:79_3"><i><font color="Green" title="E53">A22</font></i></a>, <a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:79_3"/>
<font color="Maroon" title="c5">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E2:79"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="waybel25.html#E9:79_3"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="waybel25.html#E11:79_3"><i><font color="Green" title="E55">A24</font></i></a>, <a class="ref" href="waybel_3.html#T28" title="WAYBEL_3:th.28">WAYBEL_3:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><font color="Maroon" title="c5">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c4">x</font> = 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="txt" href="waybel25.html#E7:79"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></i>
<br/>.= 
 <a href="waybel_2.html#K1" title="WAYBEL_2:func.1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" title="WAYBEL25:func.2">commute</a> <font color="Maroon" title="c3">N</font>,<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_2.html#D1" title="WAYBEL_2:def.1">WAYBEL_2:def 1</a></i>
;<br/><br/>


</div>
