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<div class="add">

<b>let </b><font color="Maroon">0L</font> be  <a href="lattices.html#V13">lower-bounded</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="lattice4.html#M2" target="_self">ClosedSubset</a> of <font color="Maroon">0L</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">0L</font>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> implies  <a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> );<br/>
<b>assume </b><a NAME="E1:26"/>
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">0L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">0L</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:26"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">B1</font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">0L</font><br/> for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">0L</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B1</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">p</font></span><a href="setwiseo.html#K2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B1</font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">0L</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">0L</font>;<br/>



<b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> implies  <a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<b>assume </b><a NAME="E2:26_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:26_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T11">XBOOLE_1:11</a></i>;<br/>
<a NAME="E4:26_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E5:26_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice4.html#K1" target="_self">FinJoin</a> <font color="Maroon">B1</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="lattice4.html#E1:26_1"><i><font color="Green">E27</font></i></a>, , , <a class="ref" href="xboole_1.html#T11">XBOOLE_1:11</a></i>;<br/>
<b>hence </b><a NAME="E6:26_1"/>
 <a href="lattice4.html#K1" target="_self">FinJoin</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">p</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:26"/>
for <font color="Olive">B</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">0L</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 <b>from </b><i><a class="ref" href="setwiseo.html#S4">SETWISEO:sch 4</a>(<a class="txt" href="lattice4.html#E2:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="lattice4.html#E3:26"><i><font color="Green">E26</font></i></a>);</i><br/>


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