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<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="lattices.html#NM4">0_Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</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="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>









<b>assume </b><a NAME="E1:68"/><i><font color="Green">E48</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>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>4</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>5</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>3</sub></font>;<br/>
<a NAME="E2:68"/><i><font color="Green">E49</font></i>: 
( the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>3</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>3</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>3</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp;  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>3</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>3</sub></font> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>3</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>, <a class="ref" href="lattice2.html#T67" target="_self">Th67</a>, <a class="ref" href="lattice2.html#T68" target="_self">Th68</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1"><b>suppose </b></a><a NAME="E1:68_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then </b> <a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lattice2.html#E2:68"><i><font color="Green">E49</font></i></a>, <a class="ref" href="setwiseo.html#T40">SETWISEO:40</a></i>
;<br/><b>hence </b><a NAME="E3:68_1_1_1"/>
 <a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#T41">LATTICES:41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_2"><b>suppose </b></a><a NAME="E1:68_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_1_1_2"/>
 <a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="lattice2.html#E1:68"><i><font color="Green">E48</font></i></a>, <a class="ref" href="lattice2.html#T47" target="_self">Th47</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:68"/>
 <a href="lattice2.html#K2" target="_self">FinJoin</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


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