<?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_0.html#NM7">sup-Semilattice</a>;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font>;<br/>
<a NAME="E1:69"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a><br/>

<b>assume </b><a NAME="E2:69"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E3:69"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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>, <font color="Maroon">t</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> such that </b><br/><a NAME="E5:69"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <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="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span> &amp; <font color="Maroon">y</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="waybel_4.html#K8" target="_self">MonSet</a> <font color="Maroon">L</font></span>)</span> &amp; <font color="Maroon">s</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">t</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:69"/>
( <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</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_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</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_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_4.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E7:69"/><b>then </b><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">AS</font> = <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>, <font color="Maroon">AT</font> = <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> as  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">AS'</font> = <font color="Maroon">AS</font>, <font color="Maroon">AT'</font> = <font color="Maroon">AT</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_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel_4.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E10:69"/>
<font color="Maroon">AS'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">AT'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<a NAME="E1:69_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AS</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AT</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:69_1_1"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AS</font>
 ;<br/>

<a NAME="E2:69_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sb</font> = <font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</font>, <font color="Maroon">tb</font> = <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b'</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> ;<br/>
<b>consider </b><font color="Maroon">a1</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> <font color="Maroon">L</font></span>)</span></span>)</span>, <font color="Maroon">b1</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> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:69_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b1</font> )
 <b>by </b><i><a class="ref" href="waybel_4.html#D4" target="_self">Def4</a>, , <a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/>
<a NAME="E7:69_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">sb</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">tb</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D6" target="_self">Def6</a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>consider </b><font color="Maroon">AR</font> being  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:69_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">AR</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">a'</font>, <font color="Olive">b'</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a'</font>,<font color="Olive">b'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR</font> iff  ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">s'</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">a'</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b'</font> &amp; <font color="Olive">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Olive">s'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y'</font> ) ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">s'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> such that </b><br/><a NAME="E11:69_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">s'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> )
 <b>by </b><i><a class="ref" href="waybel_4.html#D4" target="_self">Def4</a>, , <a class="ref" href="waybel_4.html#D8" target="_self">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon">AR1</font> being  <a href="waybel_4.html#V5" target="_self">auxiliary</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E13:69_1_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">AR1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">a'</font>, <font color="Olive">b'</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a'</font>,<font color="Olive">b'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AR1</font> iff  ex <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ex <font color="Olive">s'</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">a'</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b'</font> &amp; <font color="Olive">s'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Olive">x'</font> <a href="hidden.html#R2">in</a> <font color="Olive">s'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y'</font> ) ) ) )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E14:69_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AT</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D4" target="_self">Def4</a>, <a class="ref" href="waybel_4.html#D7" target="_self">Def7</a>, , <a class="txt" href="waybel_4.html#E7:69_1_1"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:69_1"/>
<font color="Maroon">AS'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">AT'</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:69"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">AS'</font>,<font color="Maroon">AT'</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D5" target="_self">Def5</a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:69"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K10" target="_self">Map2Rel</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_4.html#E4"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="waybel_4.html#D4" target="_self">Def4</a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div>
