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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E3:46"/><i><font color="Green">E37</font></i>: 
( the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#R6">is_distributive_wrt</a> the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V3">idempotent</a> )
 
<b>by </b><i><a class="ref" href="lattice2.html#T26" target="_self">Th26</a>, <a class="ref" href="lattice2.html#T34" target="_self">Th34</a></i>;<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>assume </b><a NAME="E1:46_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:46_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="lattice2.html#E2:46"><i><font color="Green">E36</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> = 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T60">FUNCOP_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="lattice2.html#E2:46_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>
;<br/></div><b>end;</b></div>
<span class="p1">(<span class="default"><a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">c<sub>2</sub></font> = 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>

<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>

<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funcop_1.html#K7">[:]</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice2.html#E1:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="lattice2.html#E3:46"><i><font color="Green">E37</font></i></a>, <a class="ref" href="setwiseo.html#T37">SETWISEO:37</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="lattice2.html#E1:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="lattice2.html#E3:46"><i><font color="Green">E37</font></i></a>, <a class="txt" href="lattice2.html#E4:46"><i><font color="Green">E38</font></i></a>, <a class="ref" href="setwiseo.html#T33">SETWISEO:33</a></i>
;<br/>
<b>hence </b><a NAME="E6:46"/>
 <a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>


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