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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="waybel_1.html#V8">kernel</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p1">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E1:48"/><i><font color="Green">E33</font></i>: 
 <a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel20.html#V1" target="_self">CLCongruence</a>
 
<b>by </b><i><a class="ref" href="waybel20.html#T41" target="_self">Th41</a></i>;<br/>
<a NAME="E2:48"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="waybel20.html#K4" target="_self">kernel_congruence</a> <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> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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="ref" href="waybel20.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E3:48"/><i><font color="Green">E35</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</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>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E5:48"/><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K4">dom</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> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E6:48"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="yellow_2.html#R1">&lt;=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p1">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="yellow_2.html#R1">&lt;=</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_1.html#D15">WAYBEL_1:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel20.html#E1:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel20.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E8:48"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E9:48"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relat_2.html#R8">is_transitive_in</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="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1"/>
<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>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><i><font color="Green">E40</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="quantal1.html#D9">QUANTAL1:def 9</a></i>
;<br/><a NAME="E4:48_1"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <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>
 <b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/><a NAME="E5:48_1"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/><a NAME="E6:48_1"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel20.html#E4:48"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E7:48_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel20.html#E4:48_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel20.html#E5:48_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><i><font color="Green">E44</font></i>: <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="quantal1.html#D9">QUANTAL1:def 9</a></i>
;<br/><a NAME="E9:48_1"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <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>
 <b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/><a NAME="E10:48_1"/><i><font color="Green">E46</font></i>: 
<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> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p5">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/><a NAME="E11:48_1"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel20.html#E5:48"><i><font color="Green">E37</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E12:48_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E2:48"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel20.html#E8:48_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel20.html#E9:48_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel20.html#E10:48_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E13:48_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E9:48"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel20.html#E7:48_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/><a NAME="E14:48_1"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> &amp; <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <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> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E15:48_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p5">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <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>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/><b>then </b><i><font color="Green">E48</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="quantal1.html#D9">QUANTAL1:def 9</a></i>
;<br/><a NAME="E17:48_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <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="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E6:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/><a NAME="E18:48_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E19:48_1"/>
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> &amp; <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> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <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> )
 <b>by </b><i><a class="txt" href="waybel20.html#E2:48"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel20.html#E13:48_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E20:48_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p5">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <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>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/><b>then </b><i><font color="Green">E50</font></i>: <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="quantal1.html#D9">QUANTAL1:def 9</a></i>
;<br/><a NAME="E22:48_1"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <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="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E6:48"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/><a NAME="E23:48_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E24:48_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</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>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E16:48_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="waybel20.html#E18:48_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel20.html#E21:48_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:48"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a>  <a href="waybel20.html#K3" target="_self">kernel_op</a> <span class="p1">(<span class="default"><a href="waybel20.html#K4" target="_self">kernel_congruence</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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