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

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#NM6" title="WAYBEL_0:NM.6">Semilattice</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <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> st <font color="Olive" title="b1">x</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 ;<br/>

<a NAME="E2:30_1_1"/><b>then </b><i><font color="Green" title="E17">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/>
<a NAME="E3:30_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>take </b>
<font color="Maroon" title="c2">a</font>
;<br/>

<b>let </b><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E2:30_1_1_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>hence </b><a NAME="E3:30_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:30_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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 <font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_1_1_2"/><i><font color="Green" title="E20">A4</font></i>: 
<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 <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E2:30_1_1_2"/>
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>given </b><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><a NAME="E4:30_1_1_2"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="yellow11.html#E1:30_1_1_2"><i><font color="Green" title="E20">A4</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font>;<br/>

<b>let </b><font color="Maroon" title="c7">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E6:30_1_1_2"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<div><i><font color="Green" title="E23">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_1_2_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ;<br/><a NAME="E2:30_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c4">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">a</font> )
 <b>by </b><i><a class="txt" href="yellow11.html#E4:30_1_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><b>hence </b><a NAME="E3:30_1_1_2_1"/>
<font color="Maroon" title="c4">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_1_2_2"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E2:30_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:30_1_1_2_2"/>
<font color="Maroon" title="c6">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:30_1_1_2"/>
<font color="Maroon" title="c6">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="yellow11.html#E6:30_1_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="yellow11.html#E7:30_1_1_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:30_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="yellow11.html#E2:30_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="yellow11.html#E3:30_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="yellow11.html#E4:30_1_1"><i><font color="Green" title="E19">A3</font></i></a>);</i> <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D4" title="YELLOW_0:def.4">YELLOW_0:def 4</a><br/>


</div>
