<?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#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> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/>



<a NAME="E1:8"/><i><font color="Green">E31</font></i>: 
(  <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">X</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> <font color="Maroon">X</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">I</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>  st  <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> holds <br/> <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> <font color="Olive">I</font> ) )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T61">WAYBEL_0:61</a></i>;<br/>
<a NAME="E2:8"/>
 <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> <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="E3:8"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a</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#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><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><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_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="E1:8_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:8_1"/><b>then </b>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E3:8_1"/><b>then </b>
<font color="Maroon">b</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> <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/><b>hence </b><a NAME="E4:8_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:8"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><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_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>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <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>assume </b><a NAME="E2:8_2"/>
<font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E3:8_2"/><b>then </b>
<font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><a NAME="E4:8_2"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E5:8_2"/><b>then </b>
 <a href="waybel23.html#K1" target="_self">union</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:8_2"/><b>then </b>
 <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> <font color="Maroon">b1</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T61">WAYBEL_0:61</a></i>;<br/><b>hence </b><a NAME="E7:8_2"/>
<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_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:8"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</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="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>


</div>
