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

<b>let </b><font color="Maroon">L</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">B</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">L</font>;<br/>



<a NAME="E1:100"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:100"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</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">B</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">E33</font></i>: 
 <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</font> is <a href="yellow_0.html#V6">join-inheriting</a>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">f</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">B</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">X</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">B</font></span>)</span></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">supX</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</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">B</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">unionX</font> =  <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel13.html#T3">WAYBEL13:3</a></i>;<br/><b>consider </b><font color="Maroon">J</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:100_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">supX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 <b>and </b><br/><a NAME="E5:100_1"/><i><font color="Green">E36</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">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">supX</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T3" target="_self">Th3</a></i>;<br/><b>reconsider </b><font color="Maroon">dfuX</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">X</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel13.html#T3">WAYBEL13:3</a></i>;<br/><b>reconsider </b><font color="Maroon">fuX</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">X</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</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">X</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">B</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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">L</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">E48</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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">dfuX</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">x</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">E50</font></i>: 
<font color="Maroon">x</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>consider </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:100_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">y1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E5:100_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">y1</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i>, <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">y1</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">V</font>,<font color="Maroon">L</font></span>)</span> where <font color="Olive">V</font> 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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">V</font>,<font color="Maroon">L</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D27">WAYBEL_0:def 27</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E8:100_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</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">Y</font>,<font color="Maroon">L</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> ] <b>means </b> ex <font color="Olive">I</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">B</font></span>)</span></span>)</span></span>)</span> ex <font color="Olive">z1</font>, <font color="Olive">z2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">I</font> &amp; <font color="Olive">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z2</font> );<br/>
<a NAME="E10:100_1_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> ex <font color="Olive">v</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z</font>,<font color="Olive">v</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">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:100_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">J</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">E59</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 <b>and </b><br/><a NAME="E4:100_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">J</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">I</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">I</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">B</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E7:100_1_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:100_1_1_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">J</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">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">I</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">B</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>reconsider </b><font color="Maroon">I1</font> = <font color="Maroon">I</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">B</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">B</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">I</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">L</font></span>)</span></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">J</font> = <font color="Maroon">J</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">L</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E13:100_1_1_1_1"/>
<font color="Maroon">J</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</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">z1</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">I2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E16:100_1_1_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">I1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I2</font>
 <b>and </b><br/><a NAME="E17:100_1_1_1_1"/><i><font color="Green">E117</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">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">I1</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">I2</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T3" target="_self">Th3</a></i>;<br/>
<b>consider </b><font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E19:100_1_1_1_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">z2</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z1</font>
 <b>and </b><br/><a NAME="E20:100_1_1_1_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I2</font>
 <b>by </b><i>, , , <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">z2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> <b>by </b><i>, , <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon">v</font>
;<br/>

<b>take </b>
<font color="Maroon">I</font>
;<br/>

<b>take </b>
<font color="Maroon">z1</font>
;<br/>

<b>take </b>
<font color="Maroon">z2</font>
;<br/>

<b>thus </b><a NAME="E22:100_1_1_1_1"/>
( <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z2</font> )
 <b>by </b><i><a class="ref" href="waybel23.html#T7" target="_self">Th7</a>, , <a class="ref" href="waybel23.html#T11" target="_self">Th11</a>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E12:100_1_1_1"/><i><font color="Green">E154</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>]
 <b>from </b><i><a class="ref" href="monoid_1.html#S1">MONOID_1:sch 1</a>(<a class="ref" href="waybel23.html#T6" target="_self">Th6</a>);<br/></i>
<b>reconsider </b><font color="Maroon">srg</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">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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">E156</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</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">Z</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</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">B</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">E157</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Z</font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</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">Z</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">Z</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">Z</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">Z</font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</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">Z</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">Z</font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</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">E158</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Z</font>,<font color="Maroon">L</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">Z</font>,<font color="Maroon">L</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">B</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">Z</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">Z</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">Z</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">Z</font>,<font color="Maroon">L</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">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel23.html#E8:100_1_1_1"><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">Z</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">Z</font>,<font color="Maroon">L</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">B</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">xl</font> =  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</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">B</font></span>)</span> ;<br/>
<a NAME="E20:100_1_1_1"/><i><font color="Green">E159</font></i>: 
(  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Z</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Z</font> holds <br/> <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> ) )
 
<b>by </b><i>, <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">E160</font></i>: 
<font color="Maroon">xl</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Z</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">b</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">B</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">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E3:100_1_1_1_4"/><b>then </b>
<font color="Maroon">b1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font>
 
<b>by </b><i>, <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">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">xl</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">b</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">B</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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">E161</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E3:100_1_1_1_5"/>
<font color="Maroon">b1</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Z</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">c1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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">E162</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c1</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">B</font></span>)</span> ;<br/>
<a NAME="E3:100_1_1_1_5_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="waybel23.html#T11" target="_self">Th11</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">c1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b1</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">Z</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <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">xl</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</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">E163</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font>
 
<b>by </b><i>, <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">g</font></span>)</span>,<font color="Maroon">L</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">Y</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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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">E164</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">I</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">B</font></span>)</span></span>)</span></span>)</span>, <font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">a2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:100_1_1_1_6"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:100_1_1_1_6"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:100_1_1_1_6"/>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E6:100_1_1_1_6"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E7:100_1_1_1_6"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">a1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a2</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:100_1_1_1_6"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <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">g</font></span>)</span>,<font color="Maroon">L</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</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">a2</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">g</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel23.html#T12" target="_self">Th12</a>, <a class="ref" href="waybel23.html#D2" target="_self">Def2</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">a</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">g</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel23.html#D1" target="_self">Def1</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">Y</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">srg</font>
 
<b>by </b><i><a class="ref" href="waybel23.html#T11" target="_self">Th11</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">E170</font></i>: 
<font color="Maroon">x1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">srg</font>
 
<b>by </b><i>, , <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">E171</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">X</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">X</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">g</font> <a href="tarski.html#R1">c=</a>  <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">X</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">a</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">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</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">E172</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:100_1_1_1_7"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</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">I</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">B</font></span>)</span></span>)</span></span>)</span>, <font color="Maroon">b1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:100_1_1_1_7"/>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E7:100_1_1_1_7"/>
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E8:100_1_1_1_7"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E9:100_1_1_1_7"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E10:100_1_1_1_7"/>
<font color="Maroon">b1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b2</font>
 <b>by </b><i>, <a class="ref" href="waybel23.html#T12" target="_self">Th12</a></i>;<br/>
<b>thus </b><a NAME="E11:100_1_1_1_7"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="waybel23.html#D1" target="_self">Def1</a>, <a class="ref" href="waybel23.html#D2" target="_self">Def2</a>, <a class="ref" href="waybel23.html#D3" target="_self">Def3</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">Z</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</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">V</font>,<span class="p3">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</font></span>)</span></span>)</span> where <font color="Olive">V</font> 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">X</font></span>)</span> :  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Olive">V</font>, <a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel23.html#T7" target="_self">Th7</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">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">B</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">X</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">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">dfuX</font>
 <b>by </b><i>, , <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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:100_1_1_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:100_1_1_2"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">x</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">B</font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</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">B</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/><b>consider </b><font color="Maroon">x2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:100_1_1_2"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>and </b><br/><a NAME="E6:100_1_1_2"/><i><font color="Green">E180</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">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E7:100_1_1_2"/>
<font color="Maroon">x1</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">B</font></span>)</span></span>)</span>
 <b>by </b><i>, <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">E181</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">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 <b>by </b><i>, <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">x</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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">y</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">E182</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:100_1_1_2_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">x2</font>
 
<b>by </b><i>, <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">y</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel23.html#T6" target="_self">Th6</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">X</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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">E183</font></i>: 
 <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">unionX</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><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">X</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K11">finsups</a> <font color="Maroon">unionX</font>
 <b>by </b><i>, , </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">X</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i>, <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">E184</font></i>: 
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">fuX</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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">X</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">fuX</font>
 <b>by </b><i/>;<br/><a NAME="E11:100_1_1"/><b>then </b>
<font color="Maroon">dfuX</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <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">dfuX</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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">E185</font></i>: 
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">dfuX</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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">B</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">dfuX</font>
<b>by </b><i>, , <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">B</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">X</font></span>)</span>
<b>by </b><i>, , </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">B</font></span>)</span> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X</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">B</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/>


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