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<b>let </b><font color="Maroon">P1</font> be  <a href="lattices.html#V3">strict</a> <a href="lattices.html#NM1">Lattice</a>, <font color="Maroon">P2</font> be  <a href="lattices.html#V3">strict</a> <a href="lattices.html#NM1">Lattice</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:6_1_2"/><i><font color="Green">E107</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> : <font color="Olive">x</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> </span>}</span> 
 <b>and </b><br/><a NAME="E2:6_1_2"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> holds <br/> ( the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">P1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="eqrel_1.html#K4">/\</a> <font color="Olive">y</font> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">P1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Olive">y</font> )
 <b>and </b><br/><a NAME="E3:6_1_2"/><i><font color="Green">E109</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> : <font color="Olive">x</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> </span>}</span> 
 <b>and </b><br/><a NAME="E4:6_1_2"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> holds <br/> ( the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">P2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="eqrel_1.html#K4">/\</a> <font color="Olive">y</font> &amp; the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">P2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Olive">y</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">Z</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">P1</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><i><font color="Green">E111</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/><a NAME="E1:6_1_2_1"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 ;<br/><a NAME="E2:6_1_2_1"/><b>then </b><i><font color="Green">E113</font></i>: 
 ex <font color="Olive">x2</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> &amp; <font color="Olive">x2</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><a NAME="E3:6_1_2_1"/>
 ex <font color="Olive">x3</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x3</font> &amp; <font color="Olive">x3</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>thus </b>the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">P1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">x1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">y1</font>
<b>by </b><i/>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">P2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/><a NAME="E1:6_1_2_2"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 ;<br/><a NAME="E2:6_1_2_2"/><b>then </b><i><font color="Green">E115</font></i>: 
 ex <font color="Olive">x2</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font> &amp; <font color="Olive">x2</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/><a NAME="E3:6_1_2_2"/>
 ex <font color="Olive">x3</font> being   <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x3</font> &amp; <font color="Olive">x3</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>, <font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>thus </b>the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">P1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<font color="Maroon">x1</font> <a href="eqrel_1.html#K4">/\</a> <font color="Maroon">y1</font>
<b>by </b><i/>
<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">P2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<a NAME="E8:6_1_2"/><b>then </b>
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">P1</font> <a href="funct_2.html#R1">=</a> the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">P2</font>
 
<b>by </b><i>, , <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<b>hence </b><a NAME="E9:6_1_2"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 <b>by </b><i>, , , <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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