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

<b>let </b><font color="Maroon">L</font> be   <a href="waybel_0.html#NM6">Semilattice</a>;<br/><b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:44"/><i><font color="Green">E25</font></i>: 
for <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>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; not <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> holds <br/><font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font>
 ;<br/>

<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">C</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>(  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">L</font> );<br/>
<a NAME="E2:44"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">C</font> holds  <font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E1:44_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>
<b>hence </b><a NAME="E2:44_1"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>]
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="yellow_0.html#T38">YELLOW_0:38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:44"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">B1</font>, <font color="Olive">B2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font>  st <font color="Olive">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B1</font> <a href="finsub_1.html#K1">\/</a> <font color="Olive">B2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font>, <font color="Maroon">B2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:44_2"/>
( <font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E2:44_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B2</font>] )
 ;<br/>

<a NAME="E3:44_2"/>
( <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E4:44_2"/><b>then </b>
(  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B1</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B2</font>,<font color="Maroon">L</font> or  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B2</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B1</font>,<font color="Maroon">L</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:44_2"/><b>then </b><i><font color="Green">E30</font></i>: 
( <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B1</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B2</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B1</font>,<font color="Maroon">L</font> or <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B1</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B2</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B2</font>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/>
<a NAME="E6:44_2"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B2</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B1</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">B2</font>,<font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T4">YELLOW_2:4</a></i>;<br/>
<b>hence </b><a NAME="E7:44_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B1</font> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">B2</font>]
 <b>by </b><i>, <a class="txt" href="waybel12.html#E3:44"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="yellow_2.html#T4">YELLOW_2:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:44"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font>  st <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S3">SETWISEO:sch 3</a>(<a class="txt" href="waybel12.html#E1:44"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel12.html#E2:44"><i><font color="Green">E26</font></i></a>);<br/></i>
<a NAME="E5:44"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:44"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div>
