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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">k</font> be  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="waybel_1.html#V8" title="WAYBEL_1:attr.8">kernel</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>;<br/>



<b>set </b><font color="Maroon" title="c3">kc</font> =  <a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font>;<br/>
<b>set </b><font color="Maroon" title="c4">cL</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>set </b><font color="Maroon" title="c5">km</font> =  <a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p1">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span>;<br/>
<a NAME="E1:48"/><i><font color="Green" title="E32">A1</font></i>: 
 <a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font> is <a href="waybel20.html#V1" title="WAYBEL20:attr.1">CLCongruence</a>
 
<b>by </b><i><a class="ref" href="waybel20.html#T41" target="_self" title="WAYBEL20:th.41">Th41</a></i>;<br/>
<a NAME="E2:48"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
 <a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel20.html#T37" target="_self" title="WAYBEL20:th.37">Th37</a></i>;<br/>
<a NAME="E3:48"/><i><font color="Green" title="E34">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green" title="E35">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">k</font></span>)</span></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E5:48"/><i><font color="Green" title="E36">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E6:48"/><i><font color="Green" title="E37">A6</font></i>: 
( <font color="Maroon" title="c2">k</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font> &amp;  <a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p1">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="waybel_1.html#D15" title="WAYBEL_1:def.15">WAYBEL_1:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">kc'</font> =  <a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel20.html#E1:48"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="waybel20.html#D2" target="_self" title="WAYBEL20:def.2">Def2</a></i>;<br/>
<a NAME="E8:48"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">kc'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<a NAME="E9:48"/><b>then </b><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c6">kc'</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">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" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><i><font color="Green" title="E39">A8</font></i>: <font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font>
<b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font>
<b>by </b><i><a class="ref" href="quantal1.html#D9" title="QUANTAL1:def.9">QUANTAL1:def 9</a></i>
;<br/><a NAME="E4:48_1"/><i><font color="Green" title="E40">A9</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/><a NAME="E5:48_1"/><i><font color="Green" title="E41">A10</font></i>: 
<span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><a NAME="E6:48_1"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<font color="Maroon" title="c8">x'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="waybel20.html#E4:48"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E7:48_1"/><b>then </b><i><font color="Green" title="E42">A11</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<font color="Maroon" title="c8">x'</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="waybel20.html#E4:48_1"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="waybel20.html#E5:48_1"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><i><font color="Green" title="E43">A12</font></i>: <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font>
<b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font>
<b>by </b><i><a class="ref" href="quantal1.html#D9" title="QUANTAL1:def.9">QUANTAL1:def 9</a></i>
;<br/><a NAME="E9:48_1"/><i><font color="Green" title="E44">A13</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/><a NAME="E10:48_1"/><i><font color="Green" title="E45">A14</font></i>: 
<span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">x'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p5">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><a NAME="E11:48_1"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c8">x'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="waybel20.html#E5:48"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E12:48_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c8">x'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E2:48"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="waybel20.html#E8:48_1"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="waybel20.html#E9:48_1"><i><font color="Green" title="E44">A13</font></i></a>, <a class="txt" href="waybel20.html#E10:48_1"><i><font color="Green" title="E45">A14</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E13:48_1"/><b>then </b><i><font color="Green" title="E46">A15</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E9:48"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="waybel20.html#E7:48_1"><i><font color="Green" title="E42">A11</font></i></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/><a NAME="E14:48_1"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> &amp; <span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c2">k</font></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E15:48_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p5">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><b>then </b><i><font color="Green" title="E47">A16</font></i>: <font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span> = 
<font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font>
<b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font>
<b>by </b><i><a class="ref" href="quantal1.html#D9" title="QUANTAL1:def.9">QUANTAL1:def 9</a></i>
;<br/><a NAME="E17:48_1"/>
<font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E6:48"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="yellow_2.html#T10" title="YELLOW_2:th.10">YELLOW_2:10</a></i>;<br/><a NAME="E18:48_1"/><b>then </b><i><font color="Green" title="E48">A17</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E19:48_1"/>
( <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> &amp; <span class="p1"><a href="waybel20.html#K1" title="WAYBEL20:func.1">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E2:48"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="waybel20.html#E13:48_1"><i><font color="Green" title="E46">A15</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/><a NAME="E20:48_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p4">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p5">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/><b>then </b><i><font color="Green" title="E49">A18</font></i>: <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p3">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font>
<b>by </b><i><a class="txt" href="waybel20.html#E3:48"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font>
<b>by </b><i><a class="ref" href="quantal1.html#D9" title="QUANTAL1:def.9">QUANTAL1:def 9</a></i>
;<br/><a NAME="E22:48_1"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E6:48"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="yellow_2.html#T10" title="YELLOW_2:th.10">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" title="WAYBEL20:func.3">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x'</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E24:48_1"/>
<font color="Maroon" title="c2">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p2">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E16:48_1"><i><font color="Green" title="E47">A16</font></i></a>, <a class="txt" href="waybel20.html#E18:48_1"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="waybel20.html#E21:48_1"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:48"/>
<font color="Maroon" title="c2">k</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <span class="p1">(<span class="default"><a href="waybel20.html#K4" title="WAYBEL20:func.4">kernel_congruence</a> <font color="Maroon" title="c2">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>


</div>
