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

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> ;<br/>
<b>thus </b><a NAME="E1:40_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a>  is <a href="waybel_0.html#V9">antitone</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

<i><font color="Green">E19</font></i>:  <a href="waybel_0.html#K1">netmap</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> = 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span>

<br/>.= 
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="waybel_9.html#D6" target="_self">Def6</a></i>
;<br/>
<a NAME="E2:40_1_1_1"/><i><font color="Green">E20</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel_9.html#T11" target="_self">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</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"><font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#K7">~</a> </span>)</span> ;<br/>
<a NAME="E5:40_1_1_1"/><i><font color="Green">E21</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #)
 
;<br/>
<a NAME="E6:40_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_0.html#V5">antitone</a>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T40">YELLOW_7:40</a></i>;<br/>
<a NAME="E7:40_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V5">antitone</a>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T41">YELLOW_7:41</a></i>;<br/>
<b>hence </b><a NAME="E8:40_1_1_1"/>
 <a href="waybel_0.html#K1">netmap</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V5">antitone</a>
 <b>by </b><i><a class="txt" href="waybel_9.html#E1:40_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="waybel_9.html#E2:40_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="waybel_9.html#E5:40_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="waybel_9.html#T2" target="_self">Th2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D10">WAYBEL_0:def 10</a><br/>


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

<a NAME="E2:40_1_1"/>
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"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span>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"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span>;<br/>

<b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E1:40_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel_9.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E3:40_1_1_2"/><i><font color="Green">E20</font></i>: 
( <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>;<br/>
<a NAME="E4:40_1_1_2"/>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> <a href="funct_2.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E5:40_1_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:40_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E6:40_1_1_2"/><i><font color="Green">E22</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relset_1.html#K6">~</a> 
 
<b>by </b><i><a class="ref" href="waybel_9.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E7:40_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:40_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E8:40_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> <a href="relset_1.html#K6">~</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:40_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a> </span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E5:40_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="waybel_9.html#E6:40_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="waybel_9.html#K4" target="_self">opp+id</a>  is <a href="waybel_0.html#V11">eventually-filtered</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#T12">WAYBEL_0:12</a></i>;<br/>


</div>
