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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM7" title="WAYBEL_0:NM.7">sup-Semilattice</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_3.html#D5" title="ORDERS_3:def.5">ORDERS_3:def 5</a><br/>



<a NAME="E1:66"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E2:66"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_4.html#K3" title="WAYBEL_4:func.3">Aux</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">R1</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">R2</font> = <font color="Maroon" title="c3">y</font> as  <a href="waybel_4.html#V5" title="WAYBEL_4:attr.5">auxiliary</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel_4.html#D9" target="_self" title="WAYBEL_4:def.9">Def9</a></i>;<br/>
<b>assume </b><a NAME="E4:66"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font>
 ;<br/>

<a NAME="E5:66"/><b>then </b><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">YELLOW_1:3</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" title="WAYBEL_4:func.8">MonSet</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>

<b>assume </b><a NAME="E6:66"/><i><font color="Green" title="E40">A2</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> )
 ;<br/>

<b>thus </b><a NAME="E7:66"/>
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">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" title="E41">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">R1</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a>  &amp; <span class="p1">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R2</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a>  )
 
<b>by </b><i><a class="ref" href="waybel_4.html#D15" target="_self" title="WAYBEL_4:def.15">Def15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">s</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:66_1"/><i><font color="Green" title="E42">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c8">s</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c8">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="ref" href="waybel_4.html#D14" target="_self" title="WAYBEL_4:def.14">Def14</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">t</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:66_1"/><i><font color="Green" title="E43">A5</font></i>: 
( <span class="p1">(<span class="default"><a href="waybel_4.html#K9" title="WAYBEL_4:func.9">Rel2Map</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> &amp; <font color="Maroon" title="c9">t</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds  <font color="Maroon" title="c9">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="ref" href="waybel_4.html#D14" target="_self" title="WAYBEL_4:def.14">Def14</a></i>;<br/>
<a NAME="E6:66_1"/>
<font color="Maroon" title="c8">s</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">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" title="c10">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a><br/>

<b>assume </b><a NAME="E1:66_1_1"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">q'</font> = <font color="Maroon" title="c10">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:66_1_1"/><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">R1</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c11">q'</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E1:66_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="waybel_4.html#E3:66_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="waybel_4.html#D13" target="_self" title="WAYBEL_4:def.13">Def13</a></i>;<br/>
<a NAME="E4:66_1_1"/><i><font color="Green" title="E45">A7</font></i>: 
<font color="Maroon" title="c9">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R2</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c11">q'</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E1:66_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="waybel_4.html#E5:66_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="waybel_4.html#D13" target="_self" title="WAYBEL_4:def.13">Def13</a></i>;<br/>
<a NAME="E5:66_1_1"/><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c4">R1</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c11">q'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R2</font> <a href="waybel_4.html#K5" title="WAYBEL_4:func.5">-below</a> <font color="Maroon" title="c11">q'</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E5:66"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="waybel_4.html#T29" target="_self" title="WAYBEL_4:th.29">Th29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">sq</font> = <font color="Maroon" title="c8">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">q'</font>, <font color="Maroon" title="c13">tq</font> = <font color="Maroon" title="c9">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">q'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> ;<br/>
<a NAME="E7:66_1_1"/>
<font color="Maroon" title="c12">sq</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c13">tq</font>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E3:66_1_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="waybel_4.html#E4:66_1_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="waybel_4.html#E5:66_1_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="yellow_1.html#T3" title="YELLOW_1:th.3">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" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&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" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">R1</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">R2</font> <a href="waybel_4.html#K7" title="WAYBEL_4:func.7">-below</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel_4.html#K8" title="WAYBEL_4:func.8">MonSet</a> <font color="Maroon" title="c1">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_4.html#E1:66_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="waybel_4.html#E3:66_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="waybel_4.html#E5:66_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="waybel_4.html#D14" target="_self" title="WAYBEL_4:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E8:66_1"/>
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="waybel_4.html#E6:66"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="waybel_4.html#E1:66_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


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


</div>
