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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="waybel20.html#V1" target="_self">CLCongruence</a>
 ;<br/>

<a NAME="E2:40"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel20.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:40"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel20.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:40"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="waybel_5.html#NM3">CLSubFrame</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel20.html#D2" target="_self">Def2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E5:40"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E6:40"/>
 <a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <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">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:40_1"/><i><font color="Green">E33</font></i>: 
( not <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 <b>and </b><br/><a NAME="E2:40_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E3:40_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>
<a NAME="E4:40_1"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></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">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:40_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:40_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E6:40_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E7:40_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:40_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="yellow_3.html#K7">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> ;<br/>
<a NAME="E6:40_1"/><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E7:40_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="yellow_3.html#K7">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="yellow_3.html#K7">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:40_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span><a href="yellow_3.html#K7">]</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>
<a NAME="E4:40_1_2"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel20.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E5:40_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel20.html#E3:40_1_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel20.html#T36" target="_self">Th36</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="yellow_3.html#K7">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="yellow_3.html#K7">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span> <b>by </b><i><a class="txt" href="waybel20.html#E7:40_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:40_1"/><i><font color="Green">E37</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:40_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel20.html#E12:40_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E14:40_1"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E12:40_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E15:40_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="waybel_0.html#V1">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">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E1:40_1_3"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:40_1_3"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="yellow_3.html#K7">]</a></span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:40_1_3"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="yellow_3.html#K7">]</a></span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40_1_3"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:40_1_3"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel20.html#E3:40_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel20.html#E5:40_1_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span><a href="yellow_3.html#K7">]</a></span>;<br/>
<a NAME="E8:40_1_3"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span><a href="yellow_3.html#K7">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>19</sub></font>
;<br/>

<b>thus </b><a NAME="E10:40_1_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green">E44</font></i></a></i>;<br/>

<a NAME="E11:40_1_3"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_1.html#D2">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">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<b>hence </b><a NAME="E13:40_1_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>

<a NAME="E14:40_1_3"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_1.html#D2">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">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E7:40_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<b>hence </b><a NAME="E16:40_1_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40_1_3"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow_0.html#T61">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">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_1_4_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_4_1"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:40_1_4_1"/><i><font color="Green">E42</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="yellow_3.html#K7">]</a></span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_4_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E6:40_1_4_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40_1_4_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E7:40_1_4_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel20.html#E5:40_1_4_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:40_1_4"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:40_1_4"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel20.html#E4:40_1_4"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E6:40_1_4"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E4:40_1_4"><i><font color="Green">E41</font></i></a></i>;<br/><b>hence </b><a NAME="E7:40_1_4"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E4:40_1_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:40_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">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">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_1_5_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_5_1"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:40_1_5_1"/><i><font color="Green">E43</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span><a href="yellow_3.html#K7">]</a></span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40_1_5_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>thus </b><a NAME="E6:40_1_5_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40_1_5_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:40_1_5"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E4:40_1_5"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E2:40_1_5"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E5:40_1_5"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="funct_5.html#D2">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">proj2</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E20:40_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E10:40_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel20.html#E17:40_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="waybel20.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E21:40_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:40"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel20.html#E6:40_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel20.html#E10:40_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel20.html#E14:40_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel20.html#E15:40_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel_0.html#D4">WAYBEL_0:def 4</a></i>;<br/>
<a NAME="E22:40_1"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E3:40"><i><font color="Green">E30</font></i></a>, <a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E23:40_1"/>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel20.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E24:40_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E22:40_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E25:40_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow_0.html#D3">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">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:40_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="yellow_3.html#K3">:]</a></span>
 ;<br/><a NAME="E3:40_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:40_2_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel20.html#E5:40_2_1"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E7:40_2_1"/><i><font color="Green">E35</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel20.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E8:40_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel20.html#E1:40_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel20.html#E5:40_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="eqrel_1.html#T27">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" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E7:40_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/><a NAME="E10:40_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/><a NAME="E11:40_2_1"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/><a NAME="E12:40_2_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/><a NAME="E13:40_2_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E14:40_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel20.html#E5:40_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel20.html#E10:40_2_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel20.html#E11:40_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:40_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E3:40_2"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> &amp; <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E4:40_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="yellow_3.html#K6">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:40_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel20.html#E6:40_2"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E8:40_2"/>
<span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="waybel20.html#E6:40_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/><a NAME="E9:40_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E3:40_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel20.html#E6:40_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/><a NAME="E10:40_2"/>
( <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="waybel20.html#K2" target="_self">EqRel</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel20.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E11:40_2"/><b>then </b><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E1:40"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel20.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E12:40_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E2:40"><i><font color="Green">E29</font></i></a>, <a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/><b>hence </b><a NAME="E13:40_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel20.html#E2:40"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel20.html#E6:40_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel20.html#E9:40_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel20.html#E11:40_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:40"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="waybel20.html#K1" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel20.html#K3" target="_self">kernel_op</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="waybel20.html#K1" target="_self">:]</a></span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
