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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="waybel23.html#V6" target="_self">with_bottom</a>  <a href="waybel23.html#M1" target="_self">CLbasis</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:100"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="waybel23.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E2:100"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">c<sub>1</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_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E3:100"/><i><font color="Green">E56</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font> is <a href="yellow_0.html#V6">join-inheriting</a>
 
<b>by </b><i><a class="ref" href="waybel23.html#D2" target="_self">Def2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="waybel_0.html#NM3">Ideal</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel13.html#T3">WAYBEL13:3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:100_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E5:100_1"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="waybel23.html#D11" target="_self">Def11</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel13.html#T3">WAYBEL13:3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="waybel13.html#T3">WAYBEL13:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:100_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>4</sub></font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/><b>thus </b><a NAME="E2:100_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E3:100_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_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="E1:100_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:100_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E5:100_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p4">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E1:100_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>
<a NAME="E6:100_1_1_1"/>
<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"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p4">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel23.html#E5:100_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<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 <span class="p1">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p4">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:100_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E9:100_1_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>3</sub></font> );<br/>
<a NAME="E10:100_1_1_1"/><i><font color="Green">E64</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>13</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_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> ;<br/>

<b>assume </b><a NAME="E1:100_1_1_1_1"/>
<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>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:100_1_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E4:100_1_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</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="E6:100_1_1_1_1"/>
<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="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E7:100_1_1_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E8:100_1_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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="waybel23.html#E4:100_1_1_1_1"><i><font color="Green">E66</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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel23.html#E7:100_1_1_1_1"><i><font color="Green">E67</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="waybel_0.html#NM3">Ideal</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/>
<a NAME="E11:100_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel23.html#E8:100_1_1_1_1"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E13:100_1_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel23.html#E3:100_1_1_1_1"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E16:100_1_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E17:100_1_1_1_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="waybel23.html#D11" target="_self">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E19:100_1_1_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E20:100_1_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E3:100_1_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="waybel23.html#E8:100_1_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="waybel23.html#E17:100_1_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel23.html#E16:100_1_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="waybel23.html#E20:100_1_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>20</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>22</sub></font>
;<br/>

<b>thus </b><a NAME="E22:100_1_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="waybel23.html#E7:100_1_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="waybel23.html#E16:100_1_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="waybel23.html#E19:100_1_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="waybel23.html#E20:100_1_1_1_1"><i><font color="Green">E72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E12:100_1_1_1"/><i><font color="Green">E65</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>13</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>14</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="monoid_1.html#S1">MONOID_1:sch 1</a>(<a class="txt" href="waybel23.html#E10:100_1_1_1"><i><font color="Green">E64</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<a NAME="E14:100_1_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="finset_1.html#T26">FINSET_1:26</a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E16:100_1_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>16</sub></font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_1_1_1_2_1"/>
(  not <font color="Maroon">c<sub>16</sub></font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">c<sub>16</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:100_1_1_1_2_1_1"/>
not <font color="Maroon">c<sub>16</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:100_1_1_1_2_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>16</sub></font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T54">YELLOW_0:54</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:100_1_1_1_2_1_2"/>
<font color="Maroon">c<sub>16</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:100_1_1_1_2_1_2"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>16</sub></font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T42">YELLOW_0:42</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E17:100_1_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E18:100_1_1_1"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</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_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_1_1_1_3_1"/>
(  not <font color="Maroon">c<sub>16</sub></font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">c<sub>16</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:100_1_1_1_3_1_1"/>
not <font color="Maroon">c<sub>16</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:100_1_1_1_3_1_1"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</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_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E3:100"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel21.html#T15">WAYBEL21:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:100_1_1_1_3_1_2"/>
<font color="Maroon">c<sub>16</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:100_1_1_1_3_1_2"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</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_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E2:100"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> =  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E20:100_1_1_1"/><i><font color="Green">E69</font></i>: 
(  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>16</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>16</sub></font> holds <br/> <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="waybel23.html#E17:100_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E21:100_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>16</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<b>assume </b><a NAME="E2:100_1_1_1_4"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E3:100_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E20:100_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:100_1_1_1_4"/>
<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/><b>assume </b><a NAME="E2:100_1_1_1_5"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E3:100_1_1_1_5"/>
<font color="Maroon">c<sub>19</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>16</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:100_1_1_1_5_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E3:100_1_1_1_5_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E2:100_1_1_1_5"><i><font color="Green">E71</font></i></a>, <a class="txt" href="waybel23.html#E1:100_1_1_1_5_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:100_1_1_1_5_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:100_1_1_1_5"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E17:100_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/><b>hence </b><a NAME="E5:100_1_1_1_5"/>
<font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:100_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E21:100_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E24:100_1_1_1"/>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>13</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:100_1_1_1_6"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>18</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>19</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>, <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:100_1_1_1_6"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E4:100_1_1_1_6"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>21</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>18</sub></font>
 <b>and </b><br/><a NAME="E5:100_1_1_1_6"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:100_1_1_1_6"/>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E7:100_1_1_1_6"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E12:100_1_1_1"><i><font color="Green">E65</font></i></a></i>;<br/>
<a NAME="E8:100_1_1_1_6"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E14:100_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="waybel23.html#E1:100_1_1_1_6"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:100_1_1_1_6"/>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E10:100_1_1_1_6"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E4:100_1_1_1_6"><i><font color="Green">E74</font></i></a>, <a class="txt" href="waybel23.html#E8:100_1_1_1_6"><i><font color="Green">E76</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E11:100_1_1_1_6"/>
<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E3:100_1_1_1_6"><i><font color="Green">E73</font></i></a>, <a class="txt" href="waybel23.html#E7:100_1_1_1_6"><i><font color="Green">E75</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:100_1_1_1"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E23:100_1_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E26:100_1_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E4:100_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="waybel23.html#E8:100_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E27:100_1_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a></i>;<br/>
<a NAME="E28:100_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_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="E1:100_1_1_1_7"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:100_1_1_1_7"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E4:100_1_1_1_7"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>18</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>19</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>, <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:100_1_1_1_7"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E7:100_1_1_1_7"/>
<font color="Maroon">c<sub>22</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>19</sub></font>
 <b>and </b><br/><a NAME="E8:100_1_1_1_7"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E9:100_1_1_1_7"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E10:100_1_1_1_7"/>
<font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E12:100_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="waybel23.html#E3:100_1_1_1_7"><i><font color="Green">E74</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:100_1_1_1_7"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E4:100_1_1_1_7"><i><font color="Green">E75</font></i></a>, <a class="txt" href="waybel23.html#E8:100_1_1_1_7"><i><font color="Green">E76</font></i></a>, <a class="txt" href="waybel23.html#E9:100_1_1_1_7"><i><font color="Green">E77</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:100_1_1_1"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</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 <span class="p1">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">b<sub>1</sub></font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel23.html#E16:100_1_1_1"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E30:100_1_1_1"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/>
<b>hence </b><a NAME="E31:100_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E26:100_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="waybel23.html#E27:100_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:100_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:100_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<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#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="waybel_0.html#NM3">Ideal</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel23.html#E1:100_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:100_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E6:100_1_1_2"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="waybel23.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E7:100_1_1_2"/>
<font color="Maroon">c<sub>11</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="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E2:100_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:100_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E1:100_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>thus </b><a NAME="E9:100_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel23.html#K1" target="_self">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:100_1_1_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:100_1_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="waybel23.html#E5:100_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a></i>;<br/>
<b>hence </b><a NAME="E3:100_1_1_2_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel23.html#K1" target="_self">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E6:100_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="waybel23.html#E8:100_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="waybel23.html#E1:100_1_1_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:100_1_1"/><b>then </b>
 <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel23.html#K1" target="_self">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/><a NAME="E6:100_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p4">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel23.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E7:100_1_1"/>
 <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel23.html#E2:100"><i><font color="Green">E55</font></i></a>, <a class="txt" href="waybel23.html#E3:100"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel23.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E8:100_1_1"/><b>then </b>
 <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p4">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E6:100_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:100_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_3.html#T6">YELLOW_3:6</a></i>;<br/><a NAME="E10:100_1_1"/>
 <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="waybel23.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E11:100_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E9:100_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E12:100_1_1"/><b>then </b>
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p3">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p0">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_3.html#T6">YELLOW_3:6</a></i>;<br/><a NAME="E13:100_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel23.html#T7" target="_self">Th7</a></i>;<br/><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="waybel23.html#K4" target="_self">idsMap</a> <span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><a href="waybel23.html#K1" target="_self">union</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p5">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel23.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="waybel23.html#E3:100_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="waybel23.html#E13:100_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></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>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel23.html#E4:100_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="waybel23.html#E5:100_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="waybel23.html#T6" target="_self">Th6</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:100_1"/>
 <a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:100"/>
 <a href="waybel23.html#K4" target="_self">idsMap</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="waybel_0.html#V18">sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/>


</div>
