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<b>let </b><font color="Maroon">L</font> be   <a href="waybel_0.html#NM6">Semilattice</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Olive">x</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E1:31_1_1"/>
<font color="Maroon">L</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<a NAME="E2:31_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<a NAME="E3:31_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

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

<b>assume </b><a NAME="E2:31_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>hence </b><a NAME="E3:31_1_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:31_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:31_1_1_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>

<b>given </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E3:31_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">B</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">b</font> = <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>;<br/>

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

<b>assume </b><a NAME="E5:31_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1_1_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:31_1_1_2_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><b>hence </b><a NAME="E3:31_1_1_2_1"/>
<font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1_1_2_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:31_1_1_2_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:31_1_1_2_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:31_1_1_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="yellow11.html#T1" target="_self">Th1</a>, <a class="ref" href="yellow11.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:31_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(, , );</i> <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D4">YELLOW_0:def 4</a><br/>


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