<?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>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="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>, <font color="Maroon">y</font> be   <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a><br/>



<a NAME="E1:66"/>
( <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_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font></span>)</span></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="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> )
 
;<br/>
<a NAME="E2:66"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K3" target="_self">Aux</a> <font color="Maroon">L</font> &amp; <font color="Maroon">y</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">R1</font> = <font color="Maroon">x</font>, <font color="Maroon">R2</font> = <font color="Maroon">y</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>assume </b><a NAME="E4:66"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E5:66"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>let </b><font color="Maroon">a</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">b</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>;<br/>

<b>assume </b><a NAME="E6:66"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 ;<br/>

<b>thus </b><a NAME="E7:66"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<a NAME="E1:66_1"/><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R1</font> <a href="waybel_4.html#K7" target="_self">-below</a>  &amp; <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font> <a href="waybel_4.html#K7" target="_self">-below</a>  )
 
<b>by </b><i/>;<br/>
<b>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><b> such that </b><br/><a NAME="E3:66_1"/><i><font color="Green">E34</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">x</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><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="E5:66_1"/><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K9" target="_self">Rel2Map</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">x</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E6:66_1"/>
<font color="Maroon">s</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a><br/>

<b>assume </b><a NAME="E1:66_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:66_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R1</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">q'</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>, </i>;<br/>
<a NAME="E4:66_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">q'</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#D5" target="_self">Def5</a>, </i>;<br/>
<a NAME="E5:66_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">R1</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">q'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R2</font> <a href="waybel_4.html#K5" target="_self">-below</a> <font color="Maroon">q'</font>
 
<b>by </b><i>, <a class="txt" href="waybel_4.html#E5:66_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">sq</font> = <font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q'</font>, <font color="Maroon">tq</font> = <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q'</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/>
<a NAME="E7:66_1_1"/>
<font color="Maroon">sq</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">tq</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>hence </b><a NAME="E8:66_1_1"/>
 ex <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_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:66_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R1</font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">R2</font> <a href="waybel_4.html#K7" target="_self">-below</a> </span>)</span></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="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="waybel_4.html#D5" target="_self">Def5</a>, </i>;<br/>
<b>hence </b><a NAME="E8:66_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</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="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


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


</div>
