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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="yellow_1.html#V1">RelStr-yielding</a> <a href="waybel_3.html#V4" target="_self">non-Empty</a> <a href="waybel_3.html#V5" target="_self">reflexive-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:62"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="ref" href="waybel_3.html#T31" target="_self">Th31</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:62_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E1:62_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D1" target="_self">WAYBEL_3:def 1</a><br/>

<b>assume </b><a NAME="E1:62_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>7</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>8</sub></font>
 ;<br/>

<a NAME="E2:62_1_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T27" target="_self">Th27</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span> ;<br/>
<a NAME="E3:62_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:62_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:62_1_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>
<a NAME="E4:62_1_1_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E2:62_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:62_1_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><a NAME="E2:62_1_1_1_1_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E2:62_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:62_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E6:62_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E3:62_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel_3.html#E4:62_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_3.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E7:62_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> ;<br/>
<a NAME="E7:62_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#T27" target="_self">Th27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E9:62_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> is <a href="waybel_0.html#V1">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</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="E1:62_1_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:62_1_1_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>
<b>assume </b><a NAME="E4:62_1_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:62_1_1_1_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span><b> such that </b><br/><a NAME="E8:62_1_1_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:62_1_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_3.html#E6:62_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<a NAME="E9:62_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E8:62_1_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</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="E11:62_1_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E8:62_1_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/>

<a NAME="E12:62_1_1_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T27" target="_self">Th27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:62_1_1_1_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/><a NAME="E2:62_1_1_1_2_1"/><b>then </b>
( ( <font color="Maroon">c<sub>19</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> ) or ( <font color="Maroon">c<sub>19</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> ) )
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:62_1_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_3.html#E12:62_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:62_1_1_1_2_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E8:62_1_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:62_1_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" target="_self">Th28</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:62_1_1_1_2_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/><a NAME="E2:62_1_1_1_2_2"/><b>then </b>
( ( <font color="Maroon">c<sub>19</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> ) or ( <font color="Maroon">c<sub>19</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> ) )
 <b>by </b><i><a class="txt" href="waybel_3.html#E6:62_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel_3.html#E12:62_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:62_1_1_1_2_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E8:62_1_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:62_1_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" target="_self">Th28</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:62_1_1_1_3"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:62"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E2:62_1_1_1_3"/><b>then </b><i><font color="Green">E32</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E3:62_1_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</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="E1:62_1_1_1_3_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> ;<br/>
<a NAME="E3:62_1_1_1_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E1:62_1_1_1_3_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E4:62_1_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:62_1_1_1_3_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E7:62_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:62_1_1_1_3"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_3.html#E2:62_1_1_1_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/><a NAME="E5:62_1_1_1_3"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:62"><i><font color="Green">E26</font></i></a>, <a class="ref" href="waybel_3.html#T32" target="_self">Th32</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:62_1_1_1_3_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:62_1_1_1_3_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E7:62_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:62_1_1_1_3_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:62_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:62_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel_3.html#E1:62_1_1_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_3.html#E4:62_1_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel_3.html#E5:62_1_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:62_1_1_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>14</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T28" target="_self">Th28</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E14:62_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:62_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="waybel_3.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span><b> such that </b><br/><a NAME="E16:62_1_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E14:62_1_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<a NAME="E17:62_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E2:62_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel_3.html#E16:62_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<b>hence </b><a NAME="E18:62_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E14:62_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_3.html#E16:62_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_3.html#T28" target="_self">Th28</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> </span>}</span> ;<br/><a NAME="E3:62_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:62_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> </span>}</span> 
 ;<br/>

<a NAME="E2:62_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:62_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> =  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:62_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i><a NAME="E7:62_1"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:62_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_3.html#E6:62_1"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E2:62_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><b>then 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 <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel_3.html#E7:62_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel_3.html#T27" target="_self">Th27</a></i>;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ;<br/><a NAME="E10:62_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</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="E1:62_1_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:62_1_4"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/>
<a NAME="E4:62_1_4"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E5:62_1_4"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>then </b><i><font color="Green">E32</font></i>: <font color="Maroon">c<sub>1</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel_3.html#E7:62_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:62_1_4_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> or  not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><a NAME="E2:62_1_4_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> or ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="waybel_3.html#E5:62_1_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E3:62_1_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E8:62_1_4"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E3:62_1_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel_3.html#E6:62_1_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_3.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E9:62_1_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E13:62_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/><a NAME="E15:62_1"/>
<font color="Maroon">c<sub>14</sub></font> is <a href="waybel_0.html#V1">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</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="E1:62_1_5"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:62_1_5"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 ;<br/>
<b>assume </b><a NAME="E4:62_1_5"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:62_1_5"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>18</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E8:62_1_5"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>20</sub></font>
;<br/>

<b>thus </b><a NAME="E10:62_1_5"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E11:62_1_5"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E12:62_1_5"/><b>then </b><i><font color="Green">E32</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:62_1_5_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:62"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E2:62_1_5_1"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E6:62_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_5_1_1"/>
( ( not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> ) or ( not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> &amp; not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> ) )
 </span><b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:62_1_5_1_1_1"/>
( not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> )
 ;<br/><div class="add"><a NAME="E2:62_1_5_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:62_1_5"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel_3.html#E12:62_1_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E3:62_1_5_1_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E2:62_1_5_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:62_1_5_1_1_2"/>
( not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> &amp; not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><a NAME="E2:62_1_5_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:62_1_5"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel_3.html#E12:62_1_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E3:62_1_5_1_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:62_1_5_1_1_3"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> )
 ;<br/><div class="add"><a NAME="E2:62_1_5_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:62_1_5"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel_3.html#E12:62_1_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E3:62_1_5_1_1_3"/>
<font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:62_1_5"/>
<font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" target="_self">Th28</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_5_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:62_1_5_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:62"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E2:62_1_5_2"/><b>then </b><i><font color="Green">E33</font></i>: 
(  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E6:62_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_1_5_2_1"/>
( ( not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> ) or ( not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> &amp; not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> ) or ( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> ) )
 </span><b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_5_2_1_1"><b>suppose </b></a><a NAME="E1:62_1_5_2_1_1"/>
( not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> )
 ;<br/><div class="add"><a NAME="E2:62_1_5_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E6:62_1_5"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_3.html#E12:62_1_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E3:62_1_5_2_1_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E2:62_1_5_2"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_5_2_1_2"><b>suppose </b></a><a NAME="E1:62_1_5_2_1_2"/>
( not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> &amp; not <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><div class="add"><a NAME="E2:62_1_5_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E6:62_1_5"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_3.html#E12:62_1_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E3:62_1_5_2_1_2"/>
<font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_5_2_1_3"><b>suppose </b></a><a NAME="E1:62_1_5_2_1_3"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> )
 ;<br/><div class="add"><a NAME="E2:62_1_5_2_1_3"/><b>then </b>
( <font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E6:62_1_5"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_3.html#E12:62_1_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E3:62_1_5_2_1_3"/>
<font color="Maroon">c<sub>20</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E16:62_1_5"/>
<font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" target="_self">Th28</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K6">}</a></span> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E2:62_1_6"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/><a NAME="E4:62_1_6"/><i><font color="Green">E31</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="waybel_3.html#T27" target="_self">Th27</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:62_1_6"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E6:62_1_6"/><b>then </b>
( <font color="Maroon">c<sub>18</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>15</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E2:62_1_6"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel_3.html#E4:62_1_6"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/><b>hence </b><a NAME="E7:62_1_6"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" target="_self">Th28</a></i>;<br/></div><b>end;</b></div><a NAME="E18:62_1"/><b>then </b>
<font color="Maroon">c<sub>6</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>15</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" target="_self">Th28</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E20:62_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:62_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="waybel_3.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E22:62_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_3.html#E20:62_1"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E23:62_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</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="E1:62_1_7"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:62_1_7"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> )
 ;<br/>
<b>assume </b><a NAME="E4:62_1_7"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<a NAME="E5:62_1_7"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E6:62_1_7"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#T27" target="_self">Th27</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>then </b><i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>16</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="waybel_3.html#E22:62_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_3.html#E3:62_1_7"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel_3.html#E4:62_1_7"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_3.html#E6:62_1"><i><font color="Green">E28</font></i></a></i>
;<br/>
<a NAME="E8:62_1_7"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>19</sub></font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="txt" href="waybel_3.html#E1:62"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E9:62_1_7"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel_3.html#E20:62_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel_3.html#T28" target="_self">Th28</a>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/>
<b>hence </b><a NAME="E10:62_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:62_1_7"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel_3.html#E7:62_1_7"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel_3.html#E8:62_1_7"><i><font color="Green">E35</font></i></a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>take </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font>;<br/><b>thus </b><a NAME="E25:62_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> holds <br/> not <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:62"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>given </b><font color="Maroon">c<sub>7</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E5:62"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D1" target="_self">WAYBEL_3:def 1</a><br/>

<b>assume </b><a NAME="E6:62"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</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>8</sub></font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:62_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:62_2"/><i><font color="Green">E31</font></i>: 
 <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> is <a href="waybel_0.html#V1">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>10</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="E1:62_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:62_2_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>assume </b><a NAME="E4:62_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:62_2_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel_3.html#E3:62_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel_3.html#E6:62_2_1"><i><font color="Green">E33</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:62_2_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:62_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel_3.html#E6:62_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>17</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E10:62_2_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:62_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel_3.html#E6:62_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel_3.html#E9:62_2_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_3.html#T28" target="_self">Th28</a>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<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 <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/><a NAME="E6:62_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/><a NAME="E7:62_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:62"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel_3.html#E4:62"><i><font color="Green">E27</font></i></a>, <a class="txt" href="waybel_3.html#E6:62"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel_3.html#T28" target="_self">Th28</a>, <a class="ref" href="waybel_3.html#T32" target="_self">Th32</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:62_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5" target="_self">pi</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E3:62_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_3.html#E6:62_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel_3.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:62_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E9:62_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</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="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font>;<br/><b>thus </b><a NAME="E13:62_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E11:62_2"><i><font color="Green">E34</font></i></a></i>;<br/><b>thus </b><a NAME="E14:62_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>16</sub></font>]
 <b>by </b><i><a class="txt" href="waybel_3.html#E9:62_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel_3.html#E11:62_2"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:62"/><i><font color="Green">E31</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>and </b><br/><a NAME="E10:62"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>(<a class="txt" href="waybel_3.html#E7:62"><i><font color="Green">E30</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>9</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="waybel_3.html#E9:62"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E12:62"/>
 <a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E14:62"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E15:62"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E14:62"><i><font color="Green">E33</font></i></a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:62_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:62"><i><font color="Green">E26</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_3_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> or  not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_3_1_1"><b>suppose </b></a><a NAME="E1:62_3_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:62_3_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_3.html#E10:62"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E4:62_3_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E9:62"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel_3.html#E1:62_3_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel_3.html#E3:62_3_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:62_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E14:62"><i><font color="Green">E33</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E6:62_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E7:62_3_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:62_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel_3.html#E3:62_3_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_3_1_2"><b>suppose </b></a><a NAME="E1:62_3_1_2"/>
not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:62_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_3.html#E5:62"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:62_3_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="waybel_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel_3.html#E1:62_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:62"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28" target="_self">Th28</a></i>;<br/>


</div>
