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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be  <a href="waybel25.html#V1" target="_self">monotone-convergence</a> <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="waybel_0.html#NM2">net</a> of  <a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:79"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="waybel_0.html#V10">eventually-directed</a>
 <b>and </b><br/><a NAME="E2:79"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E3:79"/><i><font color="Green">E47</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:79"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E35"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E5:79"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="waybel_9.html#R1">ex_sup_of</a>  <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></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">c<sub>9</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:79_1"/>
 <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="waybel_0.html#V10">eventually-directed</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:79"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:79_1"/>
 <a href="waybel_9.html#R1">ex_sup_of</a>  <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel25.html#T31" target="_self">Th31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:79"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel25.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E7:79"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K5">rng</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>5</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:79_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:79_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:79_2"/><i><font color="Green">E52</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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel25.html#E3:79"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel25.html#E3:79_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="waybel25.html#E35"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E34"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E7:79_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E3:79"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:79_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E2:79"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel25.html#E6:79"><i><font color="Green">E49</font></i></a>, <a class="ref" href="waybel25.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E9:79_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:79_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel25.html#E4:79_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="waybel25.html#T24" target="_self">Th24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:79"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st  <a href="relset_1.html#K5">rng</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></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">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:79_3"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> ) ) );<br/>
<a NAME="E2:79_3"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></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">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:79_3_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Olive">b<sub>1</sub></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">E53</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>take </b>
 <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></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">c<sub>10</sub></font>, <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></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">E53</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">c<sub>10</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:79_3"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="waybel25.html#E2:79_3"><i><font color="Green">E52</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T19">WAYBEL24:19</a></i>;<br/>
<a NAME="E7:79_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>13</sub></font> be   <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">c<sub>2</sub></font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:79_3_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:79_3_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:79_3_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel25.html#E3:79_3_2"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T19">WAYBEL24:19</a></i>;<br/>
<a NAME="E7:79_3_2"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>12</sub></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">c<sub>1</sub></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">c<sub>2</sub></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/>
<a NAME="E9:79_3_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>17</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>19</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:79_3_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:79_3_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="waybel25.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E2:79_3_2_1"/><i><font color="Green">E58</font></i>: 
 <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">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">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:79_3_2_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel25.html#E4:79_3"><i><font color="Green">E53</font></i></a></i>;<br/><b>then </b><i><font color="Green">E59</font></i>: <font color="Maroon">c<sub>18</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>19</sub></font> = 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_2.html#D1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>

;<br/><a NAME="E4:79_3_2_1_1_1"/>
 <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="waybel_0.html#V10">eventually-directed</a>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:79"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E5:79_3_2_1_1_1"/><b>then </b>
 <a href="waybel_9.html#R1">ex_sup_of</a>  <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel25.html#T31" target="_self">Th31</a></i>;<br/><a NAME="E6:79_3_2_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>, <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="waybel_9.html#D3">WAYBEL_9:def 3</a></i>;<br/><a NAME="E7:79_3_2_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel25.html#E3:79_3_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="waybel25.html#E35"><i><font color="Green">Lemma29</font></i></a></i>;<br/><a NAME="E8:79_3_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:79_3_2_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:79_3_2_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="waybel25.html#E2:79_3_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="waybel25.html#E3:79_3_2_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="waybel25.html#E6:79_3_2_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="yellow_4.html#T1">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">E59</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:79_3_2_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="waybel25.html#E4:79_3_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="waybel25.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E3:79_3_2_1_1_2"/>
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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> 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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel25.html#E4:79"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:79_3_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:79_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="waybel25.html#E2:79_3_2_1_1_2"><i><font color="Green">E60</font></i></a></i>;<br/><b>hence </b><a NAME="E5:79_3_2_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="waybel25.html#E4:79_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="waybel25.html#E2:79_3_2_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="waybel25.html#E1:79_3_2_1_1_2"><i><font color="Green">E59</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">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel25.html#E7:79_3_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:79_3"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E6:79"><i><font color="Green">E49</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E9:79_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="yellow14.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:79_3"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></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">c<sub>1</sub></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">c<sub>2</sub></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/>
<a NAME="E11:79_3"/><i><font color="Green">E56</font></i>: 
 <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="yellow_1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E12:79_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E8:79_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:79_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel25.html#E2:79"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel25.html#E9:79_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="waybel25.html#E11:79_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><font color="Maroon">c<sub>5</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E7:79"><i><font color="Green">E50</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 

<br/>.= 
 <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_2.html#D1">WAYBEL_2:def 1</a></i>
;<br/><br/>


</div>
