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

<b>set </b><font color="Maroon">k</font> =  <a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font>;<br/>
<a NAME="E1:46_1_1"/>
(  <a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> &amp; <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel20.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">LR</font> being  <a href="orders_2.html#V1">strict</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a><b> such that </b><br/><a NAME="E3:46_1_1"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E4:46_1_1"/><i><font color="Green">E37</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">LR</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p4">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> </span>}</span> 
 <b>and </b><br/><a NAME="E5:46_1_1"/>
for <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">LR</font>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Olive">x</font> ) holds <br/><font color="Olive">g</font> is    <a href="waybel16.html#M1">CLHomomorphism</a> of <font color="Maroon">L</font>,<font color="Maroon">LR</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">LR</font>
;<br/>

<b>thus </b><a NAME="E6:46_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">LR</font>;<br/>

<a NAME="E7:46_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">LR</font> )
 
;<br/>
<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:46_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">u</font> )
 <b>by </b><i>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:46_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">v</font> )
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font>, <font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="txt" href="waybel20.html#E4:46_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E13:46_1_1"/><i><font color="Green">E42</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">u</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">v</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel20.html#T2" target="_self">Th2</a>, </i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:46_1_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:46_1_1_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">LR</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon">u'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">v'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:46_1_1_1"/><i><font color="Green">E44</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">u'</font></span>)</span>,<span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">v'</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v'</font> )
 <b>by </b><i/>;<br/><a NAME="E5:46_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">u'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">v'</font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E7:46_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E6:46_1_1_1"/>
( <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">u'</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">v'</font></span>)</span> )
 <b>by </b><i><a class="ref" href="waybel20.html#T2" target="_self">Th2</a>, </i>;<br/><b>hence </b><a NAME="E7:46_1_1_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">y</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E7:46_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel20.html#E9:46_1_1"><i><font color="Green">E40</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E15:46_1_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">y</font>,<font color="Maroon">L</font>
 ;<br/>

<a NAME="E16:46_1_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">LR</font>
 
<b>by </b><i>, , <a class="txt" href="waybel20.html#E4:46_1_1"><i><font color="Green">E37</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E17:46_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div>
