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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>set </b><font color="Maroon">P</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>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="yellow_2.html#K2" target="_self">SupMap</a> <font color="Maroon">L</font>;<br/>
<a NAME="E1:63"/>
for <font color="Olive">x</font>, <font color="Olive">y</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>  st <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> holds <br/>for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2" target="_self">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2" target="_self">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<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_0.html#K7">Ids</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_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E2:63_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2" target="_self">SupMap</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="yellow_2.html#K2" target="_self">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">x</font>, <font color="Maroon">J</font> = <font color="Maroon">y</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E4:63_1"/><i><font color="Green">E27</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">I</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">J</font>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/>
<a NAME="E5:63_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T5" target="_self">Th5</a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E6:63_1"/>
( <span class="p1">(<span class="default"><a href="yellow_2.html#K2" target="_self">SupMap</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>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> &amp; <span class="p1">(<span class="default"><a href="yellow_2.html#K2" target="_self">SupMap</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>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">J</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:63_1"/>
<font color="Maroon">a</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#T34">YELLOW_0:34</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:63"/>
 <a href="yellow_2.html#K2" target="_self">SupMap</a> <font color="Maroon">L</font> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i><a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a></i>;<br/>


</div>
