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<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>









<b>assume </b><a NAME="E1:42"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:42"/><i><font color="Green">E18</font></i>: 
( the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> is <a href="binop_1.html#V3">idempotent</a> )
 
<b>by </b><i><a class="txt" href="lattice2.html#E1:42"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E3:42"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">B</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">B</font>,<font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="setwiseo.html#T31">SETWISEO:31</a></i>;<br/>
<b>hence </b><a NAME="E4:42"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a>  <a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">B</font>,<font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>


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