<?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_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">R</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c2">R</font> is <a href="waybel20.html#V1" title="WAYBEL20:attr.1">CLCongruence</a>
 ;<br/>

<a NAME="E2:40"/><b>then </b><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c2">R</font> is   <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="ref" href="waybel20.html#D2" target="_self" title="WAYBEL20:def.2">Def2</a></i>;<br/>
<a NAME="E3:40"/><b>then </b><i><font color="Green" title="E30">A3</font></i>: 
 <a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="waybel20.html#D1" target="_self" title="WAYBEL20:def.1">Def1</a></i>;<br/>
<a NAME="E4:40"/><i><font color="Green" title="E31">A4</font></i>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">R</font> is   <a href="waybel_5.html#NM3" title="WAYBEL_5:NM.3">CLSubFrame</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="waybel20.html#D2" target="_self" title="WAYBEL20:def.2">Def2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">k</font> =  <a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</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/>
<a NAME="E5:40"/><i><font color="Green" title="E32">A5</font></i>: 
 <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> <font color="Maroon" title="c2">R</font></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/>
<b>thus </b><a NAME="E6:40"/>
 <a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">D</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_1"/><i><font color="Green" title="E33">A6</font></i>: 
( not <font color="Maroon" title="c5">D</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">D</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 <b>and </b><br/><a NAME="E2:40_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">L</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E3:40_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>

<b>set </b><font color="Maroon" title="c6">d</font> =  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font>;<br/>
<b>set </b><font color="Maroon" title="c7">ds</font> =  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font></span>)</span>;<br/>
<a NAME="E4:40_1"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:40_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1"/><i><font color="Green" title="E34">A7</font></i>: 
( <font color="Maroon" title="c9">a</font> <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">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c8">b</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c9">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_1"><i><font color="Green" title="E34">A7</font></i></a></i>;<br/>
<a NAME="E5:40_1_1"/>
<font color="Maroon" title="c5">D</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>
<a NAME="E6:40_1_1"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E7:40_1_1"/>
<font color="Maroon" title="c8">b</font> <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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:40_1"/><b>then </b><i><font color="Green" title="E34">A8</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32" title="YELLOW_0:th.32">YELLOW_0:32</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">S</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> </span>}</span> ;<br/>
<a NAME="E6:40_1"/><i><font color="Green" title="E35">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E7:40_1"/><i><font color="Green" title="E36">A10</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:40_1_2"/><i><font color="Green" title="E37">A11</font></i>: 
( <font color="Maroon" title="c9">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"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span>,<font color="Maroon" title="c10">a</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )
 ;<br/>
<a NAME="E4:40_1_2"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c10">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="waybel20.html#D3" target="_self" title="WAYBEL20:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E5:40_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="waybel20.html#E3:40_1_2"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="waybel20.html#T36" target="_self" title="WAYBEL20:th.36">Th36</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">S'</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">R</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">S</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<font color="Olive" title="b1">x</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span> <b>by </b><i><a class="txt" href="waybel20.html#E7:40_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:40_1"/><i><font color="Green" title="E37">A12</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c10">S</font>,<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:40_1"/><i><font color="Green" title="E38">A13</font></i>: 
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">e</font> = <font color="Maroon" title="c11">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel20.html#E12:40_1"><i><font color="Green" title="E38">A13</font></i></a></i>;<br/>
<a NAME="E14:40_1"/><i><font color="Green" title="E39">A14</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">e</font></span>)</span>,<font color="Maroon" title="c12">e</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> <font color="Maroon" title="c10">S</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E12:40_1"><i><font color="Green" title="E38">A13</font></i></a></i>;<br/>
<a NAME="E15:40_1"/><i><font color="Green" title="E40">A15</font></i>: 
<font color="Maroon" title="c9">S'</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">R</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E1:40_1_3"/><i><font color="Green" title="E41">A16</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S'</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S'</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:40_1_3"/><i><font color="Green" title="E42">A17</font></i>: 
( <font color="Maroon" title="c13">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"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span>,<font color="Maroon" title="c15">a</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> &amp; <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c16">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:40_1_3"/><i><font color="Green" title="E43">A18</font></i>: 
( <font color="Maroon" title="c14">y</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"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b</font></span>)</span>,<font color="Maroon" title="c16">b</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> &amp; <font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40_1_3"><i><font color="Green" title="E41">A16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E7:40_1_3"/><i><font color="Green" title="E44">A19</font></i>: 
( <font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c15">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">c</font> &amp; <font color="Maroon" title="c16">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">c</font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="waybel20.html#E3:40_1_3"><i><font color="Green" title="E42">A17</font></i></a>, <a class="txt" href="waybel20.html#E5:40_1_3"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c18">z</font> = <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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">c</font></span>)</span>,<font color="Maroon" title="c17">c</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>;<br/>
<a NAME="E8:40_1_3"/>
<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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">c</font></span>)</span>,<font color="Maroon" title="c17">c</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> <font color="Maroon" title="c9">S'</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green" title="E44">A19</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">z</font> = <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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">c</font></span>)</span>,<font color="Maroon" title="c17">c</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c2">R</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c19">z</font>
;<br/>

<b>thus </b><a NAME="E10:40_1_3"/>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">S'</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green" title="E44">A19</font></i></a></i>;<br/>

<a NAME="E11:40_1_3"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a</font> <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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">c</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green" title="E44">A19</font></i></a>, <a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></i>;<br/>
<a NAME="E12:40_1_3"/><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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span>,<font color="Maroon" title="c15">a</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">c</font></span>)</span>,<font color="Maroon" title="c17">c</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green" title="E44">A19</font></i></a>, <a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/>
<b>hence </b><a NAME="E13:40_1_3"/>
<font color="Maroon" title="c13">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">z</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_3"><i><font color="Green" title="E42">A17</font></i></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>;<br/>

<a NAME="E14:40_1_3"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b</font> <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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">c</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green" title="E44">A19</font></i></a>, <a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></i>;<br/>
<a NAME="E15:40_1_3"/><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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">b</font></span>)</span>,<font color="Maroon" title="c16">b</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">c</font></span>)</span>,<font color="Maroon" title="c17">c</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green" title="E44">A19</font></i></a>, <a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/>
<b>hence </b><a NAME="E16:40_1_3"/>
<font color="Maroon" title="c14">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">z</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40_1_3"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_1_4_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c10">S</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:40_1_4_1"/><i><font color="Green" title="E41">A20</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font>
 <b>by </b><i><a class="ref" href="funct_5.html#D1" title="FUNCT_5:def.1">FUNCT_5:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:40_1_4_1"/><i><font color="Green" title="E42">A21</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span>,<font color="Maroon" title="c15">a</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> &amp; <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_4_1"><i><font color="Green" title="E41">A20</font></i></a></i>;<br/><a NAME="E6:40_1_4_1"/>
<font color="Maroon" title="c13">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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40_1_4_1"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E7:40_1_4_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="waybel20.html#E5:40_1_4_1"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:40_1_4"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:40_1_4"/><i><font color="Green" title="E41">A22</font></i>: 
( <font color="Maroon" title="c14">y</font> <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">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c13">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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel20.html#E4:40_1_4"><i><font color="Green" title="E41">A22</font></i></a></i>;<br/><a NAME="E6:40_1_4"/>
<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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font></span>)</span>,<font color="Maroon" title="c15">y</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> <font color="Maroon" title="c10">S</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E4:40_1_4"><i><font color="Green" title="E41">A22</font></i></a></i>;<br/><b>hence </b><a NAME="E7:40_1_4"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c10">S</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E4:40_1_4"><i><font color="Green" title="E41">A22</font></i></a>, <a class="ref" href="funct_5.html#D1" title="FUNCT_5:def.1">FUNCT_5:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:40_1"/><b>then </b><i><font color="Green" title="E41">A23</font></i>: 
 <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c10">S</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_1_5_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c10">S</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:40_1_5_1"/><i><font color="Green" title="E42">A24</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c13">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">S</font>
 <b>by </b><i><a class="ref" href="funct_5.html#D2" title="FUNCT_5:def.2">FUNCT_5:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:40_1_5_1"/><i><font color="Green" title="E43">A25</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c13">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span>,<font color="Maroon" title="c15">a</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> &amp; <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_5_1"><i><font color="Green" title="E42">A24</font></i></a></i>;<br/><b>thus </b><a NAME="E6:40_1_5_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40_1_5_1"><i><font color="Green" title="E43">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:40_1_5"/><i><font color="Green" title="E42">A26</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c14">x'</font> = <font color="Maroon" title="c13">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/><a NAME="E4:40_1_5"/>
<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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font></span>)</span>,<font color="Maroon" title="c14">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> <font color="Maroon" title="c10">S</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E2:40_1_5"><i><font color="Green" title="E42">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E5:40_1_5"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c10">S</font>
 <b>by </b><i><a class="ref" href="funct_5.html#D2" title="FUNCT_5:def.2">FUNCT_5:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:40_1"/><b>then </b>
 <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c10">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E20:40_1"/><b>then </b>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c10">S</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"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</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#E10:40_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="waybel20.html#E17:40_1"><i><font color="Green" title="E41">A23</font></i></a>, <a class="ref" href="waybel20.html#T8" target="_self" title="WAYBEL20:th.8">Th8</a></i>;<br/>
<a NAME="E21:40_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"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</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> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:40"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="waybel20.html#E6:40_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="waybel20.html#E10:40_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="waybel20.html#E14:40_1"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="waybel20.html#E15:40_1"><i><font color="Green" title="E40">A15</font></i></a>, <a class="ref" href="waybel_0.html#D4" title="WAYBEL_0:def.4">WAYBEL_0:def 4</a></i>;<br/>
<a NAME="E22:40_1"/><b>then </b><i><font color="Green" title="E42">A27</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E3:40"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<a NAME="E23:40_1"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="waybel20.html#D3" target="_self" title="WAYBEL20:def.3">Def3</a></i>;<br/>
<a NAME="E24:40_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E22:40_1"><i><font color="Green" title="E42">A27</font></i></a>, <a class="ref" href="yellow_2.html#T24" title="YELLOW_2:th.24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E25:40_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">D</font></span>)</span> <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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c5">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40_1"><i><font color="Green" title="E34">A8</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>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_2_1"/><i><font color="Green" title="E33">A28</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 ;<br/><a NAME="E2:40_2_1"/><b>then </b>
<font color="Maroon" title="c5">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 <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 ;<br/><a NAME="E3:40_2_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:40_2_1"/><i><font color="Green" title="E34">A29</font></i>: 
( <font color="Maroon" title="c6">x1</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> &amp; <font color="Maroon" title="c7">x2</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> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c9">x2</font> = <font color="Maroon" title="c7">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel20.html#E5:40_2_1"><i><font color="Green" title="E34">A29</font></i></a></i>;<br/><a NAME="E7:40_2_1"/><i><font color="Green" title="E35">A30</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c8">x1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c9">x2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="waybel20.html#D3" target="_self" title="WAYBEL20:def.3">Def3</a></i>;<br/><a NAME="E8:40_2_1"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c9">x2</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="waybel20.html#E1:40_2_1"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="waybel20.html#E5:40_2_1"><i><font color="Green" title="E34">A29</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/><a NAME="E9:40_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x2</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E7:40_2_1"><i><font color="Green" title="E35">A30</font></i></a>, <a class="ref" href="eqrel_1.html#T31" title="EQREL_1:th.31">EQREL_1:31</a></i>;<br/><a NAME="E10:40_2_1"/><b>then </b><i><font color="Green" title="E36">A31</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x2</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="E11:40_2_1"/><i><font color="Green" title="E37">A32</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> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></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">x1</font>,<font color="Maroon" title="c9">x2</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"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x2</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#E5:40"><i><font color="Green" title="E32">A5</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="E12:40_2_1"/>
 <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> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></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> <font color="Maroon" title="c2">R</font></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> <font color="Maroon" title="c2">R</font></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="E13:40_2_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">x1</font>,<font color="Maroon" title="c9">x2</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"><span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E14:40_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></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="txt" href="waybel20.html#E5:40_2_1"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="waybel20.html#E10:40_2_1"><i><font color="Green" title="E36">A31</font></i></a>, <a class="txt" href="waybel20.html#E11:40_2_1"><i><font color="Green" title="E37">A32</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:40_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></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>
 ;<br/><a NAME="E3:40_2"/><b>then </b><i><font color="Green" title="E33">A33</font></i>: 
( <font color="Maroon" title="c5">x</font> <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> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></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> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span></span><a href="waybel20.html#K1" title="WAYBEL20:func.1">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <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="E4:40_2"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c2">R</font></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> <font color="Maroon" title="c2">R</font></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/><b>then consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:40_2"/><i><font color="Green" title="E34">A34</font></i>: 
( <font color="Maroon" title="c6">x1</font> <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">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c7">x2</font> <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">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c9">x2</font> = <font color="Maroon" title="c7">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel20.html#E6:40_2"><i><font color="Green" title="E34">A34</font></i></a></i>;<br/><a NAME="E8:40_2"/>
<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> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></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">x1</font>,<font color="Maroon" title="c9">x2</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"><span class="p3">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x2</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#E6:40_2"><i><font color="Green" title="E34">A34</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="E9:40_2"/><b>then </b><i><font color="Green" title="E35">A35</font></i>: 
<span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x2</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40_2"><i><font color="Green" title="E33">A33</font></i></a>, <a class="txt" href="waybel20.html#E6:40_2"><i><font color="Green" title="E34">A34</font></i></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/><a NAME="E10:40_2"/>
( <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c8">x1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7" title="EQREL_1:func.7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" title="WAYBEL20:func.2">EqRel</a> <font color="Maroon" title="c2">R</font></span>)</span>,<font color="Maroon" title="c9">x2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="waybel20.html#D3" target="_self" title="WAYBEL20:def.3">Def3</a></i>;<br/><a NAME="E11:40_2"/><b>then </b><i><font color="Green" title="E36">A36</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</font></span>)</span>,<font color="Maroon" title="c8">x1</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> <font color="Maroon" title="c2">R</font> &amp; <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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x2</font></span>)</span>,<font color="Maroon" title="c9">x2</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> <font color="Maroon" title="c2">R</font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="waybel20.html#T36" target="_self" title="WAYBEL20:th.36">Th36</a></i>;<br/><a NAME="E12:40_2"/><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">x1</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> <font color="Maroon" title="c2">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x1</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> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="eqrel_1.html#T12" title="EQREL_1:th.12">EQREL_1:12</a></i>;<br/><b>hence </b><a NAME="E13:40_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="waybel20.html#E2:40"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="waybel20.html#E6:40_2"><i><font color="Green" title="E34">A34</font></i></a>, <a class="txt" href="waybel20.html#E9:40_2"><i><font color="Green" title="E35">A35</font></i></a>, <a class="txt" href="waybel20.html#E11:40_2"><i><font color="Green" title="E36">A36</font></i></a>, <a class="ref" href="eqrel_1.html#T13" title="EQREL_1:th.13">EQREL_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:40"/>
<font color="Maroon" title="c2">R</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> <font color="Maroon" title="c2">R</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" title="WAYBEL20:func.3">kernel_op</a> <font color="Maroon" title="c2">R</font></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="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div>
