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

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:46_1_1"/>
(  <a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> &amp; <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:46"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel20.html#T37" target="_self">Th37</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></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">E33</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:46_1_1"/><i><font color="Green">E34</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 <b>and </b><br/><a NAME="E5:46_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font> ) holds <br/><font color="Olive">b<sub>1</sub></font> is    <a href="waybel16.html#M1">CLHomomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="waybel20.html#T38" target="_self">Th38</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E6:46_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:46_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>

<a NAME="E7:46_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:46_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E3:46_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:46_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E3:46_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel20.html#E7:46_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</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="txt" href="waybel20.html#E9:46_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel20.html#E11:46_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E13:46_1_1"/><i><font color="Green">E38</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:46"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel20.html#D3" target="_self">Def3</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:46_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></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">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:46_1_1_1"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E4:46_1_1"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E5:46_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E4:46_1_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">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel20.html#E1:46"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel20.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E7:46_1_1_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E4:46_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel20.html#E5:46_1_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">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:46_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel20.html#E9:46_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel20.html#E11:46_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel20.html#E13:46_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:46_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div>
