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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a><br/>







<b>assume </b><a NAME="E1:48"/>
( not <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</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="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<a NAME="E3:48"/>
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>3</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</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"><font color="Maroon">c<sub>3</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_2.html#T7">YELLOW_2:7</a></i>;<br/>
<a NAME="E5:48"/><i><font color="Green">E34</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<b>assume </b><a NAME="E6:48"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>7</sub></font>, <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E7:48"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font>, <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E8:48"/><i><font color="Green">E35</font></i>: 
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>4</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E9:48"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</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"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel27.html#E8:48"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<a NAME="E12:48"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E13:48"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E14:48"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#T27">WAYBEL_3:27</a></i>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1"/>
<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 <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E3:48_1"/><i><font color="Green">E40</font></i>: 
 <a href="yellow16.html#K1">pi</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K5">pi</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_1_1"/>
 <a href="yellow16.html#K1">pi</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K5">pi</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</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="E1:48_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow16.html#K1">pi</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:48_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:48_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <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>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="waybel27.html#E3:48_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel27.html#E5:48_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel27.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E7:48_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:48_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel27.html#E5:48_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<a NAME="E9:48_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:48_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:48_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
;<br/>
<a NAME="E12:48_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel27.html#E9:48_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="waybel27.html#E3:48_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel27.html#E5:48_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel27.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E12:48_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E9:48_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E14:48_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K5">pi</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel27.html#E7:48_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel27.html#E8:48_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>14</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="E2:48_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K5">pi</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:48_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp; <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_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:48_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel27.html#E4:48_1_1"><i><font color="Green">E41</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>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel27.html#E6:48_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel27.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E8:48_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:48_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="waybel27.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E10:48_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</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="waybel27.html#E6:48_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel27.html#E8:48_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E11:48_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:48_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:48_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>17</sub></font>
 
;<br/>
<a NAME="E14:48_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel27.html#E11:48_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E4:48_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel27.html#E6:48_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel27.html#E11:48_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="waybel27.html#E14:48_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
;<br/>
<b>hence </b><a NAME="E16:48_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow16.html#K1">pi</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="waybel27.html#E10:48_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:48_1"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="yellow16.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E5:48_1"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow16.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <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> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel27.html#E12:48"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E7:48_1"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="yellow16.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E8:48_1"/>
 <a href="waybel_3.html#K5">pi</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 <b>by </b><i><a class="ref" href="yellow16.html#T37">YELLOW16:37</a></i>;<br/><a NAME="E9:48_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="yellow16.html#K1">pi</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="yellow16.html#K1">pi</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="waybel27.html#E7:48_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>8</sub></font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E11:48_1"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E12:48_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E13:48_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><a NAME="E14:48_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel27.html#E11:48_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E15:48_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>10</sub></font>, <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="yellow16.html#K1">pi</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E13:48"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel27.html#E4:48_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="yellow16.html#T35">YELLOW16:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="yellow16.html#K1">pi</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E3:48_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel27.html#E9:48_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="waybel27.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E5:48"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel27.html#E12:48"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel27.html#E5:48_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="yellow16.html#T35">YELLOW16:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel27.html#T27" target="_self">Th27</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E11:48_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="waybel27.html#E14:48_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="waybel27.html#D5" target="_self">Def5</a></i>
;<br/><br/></div><b>end;</b></div>
<b>thus </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="waybel27.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E14:48"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel27.html#E15:48"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
;<br/>


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