:: LATTICE2 semantic presentation

begin

theorem :: LATTICE2:1
for A being ( ( ) ( ) set )
for C being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Subset of ( ( ) ( ) set ) )
for g being ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) holds dom (g : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | B : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) -defined b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = B : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ;

theorem :: LATTICE2:2
for A being ( ( ) ( ) set )
for C being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Subset of ( ( ) ( ) set ) )
for f, g being ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) holds
( f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | B : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) -defined b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | B : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) -defined b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) iff for x being ( ( ) ( ) Element of A : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) in B : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) holds
g : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: LATTICE2:3
for A being ( ( ) ( ) set )
for C being ( ( non empty ) ( non empty ) set )
for f, g being ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) )
for B being ( ( ) ( ) set ) holds f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) +* (g : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | B : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b5 : ( ( ) ( ) set ) -defined b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: LATTICE2:4
for A being ( ( ) ( ) set )
for C being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Subset of ( ( ) ( ) set ) )
for g, f being ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) holds (g : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | B : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) -defined b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) +* f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: LATTICE2:5
for f, g being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) st g : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) c= f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) holds
f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) +* g : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) = f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ;

theorem :: LATTICE2:6
for A being ( ( ) ( ) set )
for C being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) holds f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) +* (f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | B : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) -defined b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: LATTICE2:7
for A being ( ( ) ( ) set )
for C being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Subset of ( ( ) ( ) set ) )
for g, f being ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) Element of A : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) in B : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) holds
g : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) +* (g : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | B : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b3 : ( ( ) ( ) Subset of ( ( ) ( ) set ) ) -defined b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: LATTICE2:8
for A being ( ( ) ( ) set )
for C being ( ( non empty ) ( non empty ) set )
for g, f being ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds (g : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b5 : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) -defined b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) +* f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: LATTICE2:9
for A being ( ( ) ( ) set )
for C being ( ( non empty ) ( non empty ) set )
for g being ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds dom (g : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b4 : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) -defined b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ;

theorem :: LATTICE2:10
for A being ( ( ) ( ) set )
for C being ( ( non empty ) ( non empty ) set )
for g, f being ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A : ( ( ) ( ) set ) ,C : ( ( non empty ) ( non empty ) set ) )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) st ( for x being ( ( ) ( ) Element of A : ( ( ) ( ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
g : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) +* (g : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b5 : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) -defined b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = f : ( ( Function-like V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( ) ( ) set ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(b1 : ( ( ) ( ) set ) ) V18(b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b1 : ( ( ) ( ) set ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ;

definition
let D be ( ( non empty ) ( non empty ) set ) ;
let o, o9 be ( ( Function-like V18(K20(D : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like K20(D : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined D : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(K20(D : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20(D : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) ) BinOp of D : ( ( non empty ) ( non empty ) set ) ) ;
pred o absorbs o9 means :: LATTICE2:def 1
 for x, y being ( ( ) ( ) Element of D : ( ( ) ( ) LattStr ) ) holds o : ( ( Function-like V18(K20(D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) ) : ( ( ) ( Relation-like ) set ) ,D : ( ( ) ( ) LattStr ) ) ) ( Relation-like K20(D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) ) : ( ( ) ( Relation-like ) set ) -defined D : ( ( ) ( ) LattStr ) -valued Function-like V18(K20(D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) ) : ( ( ) ( Relation-like ) set ) ,D : ( ( ) ( ) LattStr ) ) ) Element of K19(K20(K20(D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) ) : ( ( ) ( Relation-like ) set ) ,D : ( ( ) ( ) LattStr ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ,(o9 : ( ( Function-like V18(K20(D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) ) : ( ( ) ( Relation-like ) set ) ,D : ( ( ) ( ) LattStr ) ) ) ( Relation-like K20(D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) ) : ( ( ) ( Relation-like ) set ) -defined D : ( ( ) ( ) LattStr ) -valued Function-like V18(K20(D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) ) : ( ( ) ( Relation-like ) set ) ,D : ( ( ) ( ) LattStr ) ) ) Element of K19(K20(K20(D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) ) : ( ( ) ( Relation-like ) set ) ,D : ( ( ) ( ) LattStr ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . (x : ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ,y : ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of D : ( ( ) ( ) LattStr ) ) ) : ( ( ) ( ) Element of D : ( ( ) ( ) LattStr ) ) = x : ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) ;
end;

notation
let D be ( ( non empty ) ( non empty ) set ) ;
let o, o9 be ( ( Function-like V18(K20(D : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like K20(D : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined D : ( ( non empty ) ( non empty ) set ) -valued Function-like V14(K20(D : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20(D : ( ( non empty ) ( non empty ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ,D : ( ( non empty ) ( non empty ) set ) ) ) BinOp of D : ( ( non empty ) ( non empty ) set ) ) ;
antonym o doesn't_absorb o9 for o absorbs o9;
end;

theorem :: LATTICE2:11
for L being ( ( non empty ) ( non empty ) LattStr ) st the L_join of L : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is commutative & the L_join of L : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is associative & the L_meet of L : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is commutative & the L_meet of L : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is associative & the L_join of L : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) absorbs the L_meet of L : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) & the L_meet of L : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) absorbs the L_join of L : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) holds
L : ( ( non empty ) ( non empty ) LattStr ) is Lattice-like ;

definition
let L be ( ( ) ( ) LattStr ) ;
func L .: -> ( ( strict ) ( strict ) LattStr ) equals :: LATTICE2:def 2
 LattStr(# the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the L_meet of L : ( ( ) ( ) LattStr ) : ( ( Function-like V18(K20( the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) -valued Function-like V18(K20( the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) Element of K19(K20(K20( the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) , the L_join of L : ( ( ) ( ) LattStr ) : ( ( Function-like V18(K20( the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) -valued Function-like V18(K20( the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) Element of K19(K20(K20( the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) ;
end;

registration
let L be ( ( non empty ) ( non empty ) LattStr ) ;
cluster L : ( ( non empty ) ( non empty ) LattStr ) .: : ( ( strict ) ( strict ) LattStr ) -> non empty strict ;
end;

theorem :: LATTICE2:12
for L being ( ( non empty ) ( non empty ) LattStr ) holds
( the carrier of L : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) = the carrier of (L : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) & the L_join of L : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = the L_meet of (L : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( Function-like V18(K20( the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) & the L_meet of L : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = the L_join of (L : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( Function-like V18(K20( the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of (b1 : ( ( non empty ) ( non empty ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: LATTICE2:13
for L being ( ( non empty strict ) ( non empty strict ) LattStr ) holds (L : ( ( non empty strict ) ( non empty strict ) LattStr ) .:) : ( ( strict ) ( non empty strict ) LattStr ) .: : ( ( strict ) ( non empty strict ) LattStr ) = L : ( ( non empty strict ) ( non empty strict ) LattStr ) ;

theorem :: LATTICE2:14
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for v being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) holds
u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = Bottom L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:15
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for v being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . (u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) holds
u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = Bottom L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:16
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for v being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) holds
u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = Top L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:17
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for v being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . (u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) holds
u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = Top L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ;

registration
let L be ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ;
cluster the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( Function-like V18(K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) -> Function-like V18(K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) idempotent ;
end;

registration
let L be ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) ;
cluster the L_join of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( Function-like V18(K20( the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) -> Function-like V18(K20( the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty join-commutative ) ( non empty join-commutative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) commutative ;
end;

theorem :: LATTICE2:18
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) st the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is having_a_unity holds
Bottom L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = the_unity_wrt the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ;

registration
let L be ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) ;
cluster the L_join of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( Function-like V18(K20( the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) -> Function-like V18(K20( the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty join-associative ) ( non empty join-associative ) \/-SemiLattStr ) : ( ( ) ( non empty ) set ) ) associative ;
end;

registration
let L be ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ;
cluster the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( Function-like V18(K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) -> Function-like V18(K20( the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) idempotent ;
end;

registration
let L be ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) ;
cluster the L_meet of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( Function-like V18(K20( the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) -> Function-like V18(K20( the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty meet-commutative ) ( non empty meet-commutative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) commutative ;
end;

registration
let L be ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) ;
cluster the L_meet of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( Function-like V18(K20( the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(K20( the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) -> Function-like V18(K20( the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty meet-associative ) ( non empty meet-associative ) /\-SemiLattStr ) : ( ( ) ( non empty ) set ) ) associative ;
end;

theorem :: LATTICE2:19
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) st the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is having_a_unity holds
Top L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = the_unity_wrt the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:20
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) holds the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is_distributive_wrt the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: LATTICE2:21
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) st L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) is ( ( non empty Lattice-like distributive ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular ) D_Lattice) holds
the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is_distributive_wrt the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: LATTICE2:22
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) st the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is_distributive_wrt the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) holds
L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) is distributive ;

theorem :: LATTICE2:23
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) st L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) is ( ( non empty Lattice-like distributive ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular ) D_Lattice) holds
the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is_distributive_wrt the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: LATTICE2:24
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) st the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is_distributive_wrt the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) holds
L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) is distributive ;

theorem :: LATTICE2:25
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) holds the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is_distributive_wrt the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: LATTICE2:26
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) holds the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) absorbs the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: LATTICE2:27
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) holds the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) absorbs the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;

definition
let A be ( ( non empty ) ( non empty ) set ) ;
let L be ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ;
let B be ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ;
let f be ( ( Function-like V18(A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like A : ( ( non empty ) ( non empty ) set ) -defined the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(A : ( ( non empty ) ( non empty ) set ) ) V18(A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ;
func FinJoin (B,f) -> ( ( ) ( ) Element of ( ( ) ( ) set ) ) equals :: LATTICE2:def 3
 the L_join of L : ( ( ) ( ) set ) : ( ( Function-like V18(K20( the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like V18(K20( the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) Element of K19(K20(K20( the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) $$ (B : ( ( Function-like V18(K20(A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ,A : ( ( ) ( ) set ) ) ) ( Relation-like K20(A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) -defined A : ( ( ) ( ) set ) -valued Function-like V18(K20(A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ,A : ( ( ) ( ) set ) ) ) Element of K19(K20(K20(A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( Function-like V18(A : ( ( ) ( ) set ) , Fin L : ( ( ) ( ) set ) : ( ( preBoolean ) ( preBoolean ) set ) ) ) ( Relation-like A : ( ( ) ( ) set ) -defined Fin L : ( ( ) ( ) set ) : ( ( preBoolean ) ( preBoolean ) set ) -valued Function-like V18(A : ( ( ) ( ) set ) , Fin L : ( ( ) ( ) set ) : ( ( preBoolean ) ( preBoolean ) set ) ) ) Element of K19(K20(A : ( ( ) ( ) set ) ,(Fin L : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( preBoolean ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
func FinMeet (B,f) -> ( ( ) ( ) Element of ( ( ) ( ) set ) ) equals :: LATTICE2:def 4
 the L_meet of L : ( ( ) ( ) set ) : ( ( Function-like V18(K20( the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like V18(K20( the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) Element of K19(K20(K20( the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) $$ (B : ( ( Function-like V18(K20(A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ,A : ( ( ) ( ) set ) ) ) ( Relation-like K20(A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) -defined A : ( ( ) ( ) set ) -valued Function-like V18(K20(A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ,A : ( ( ) ( ) set ) ) ) Element of K19(K20(K20(A : ( ( ) ( ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ,A : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( Function-like V18(A : ( ( ) ( ) set ) , Fin L : ( ( ) ( ) set ) : ( ( preBoolean ) ( preBoolean ) set ) ) ) ( Relation-like A : ( ( ) ( ) set ) -defined Fin L : ( ( ) ( ) set ) : ( ( preBoolean ) ( preBoolean ) set ) -valued Function-like V18(A : ( ( ) ( ) set ) , Fin L : ( ( ) ( ) set ) : ( ( preBoolean ) ( preBoolean ) set ) ) ) Element of K19(K20(A : ( ( ) ( ) set ) ,(Fin L : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( preBoolean ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ;
end;

theorem :: LATTICE2:28
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) [= FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:29
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st ex x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st
( x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) & u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) holds
u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:30
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) & B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) holds
FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:31
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) [= u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:32
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) & ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) [= u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) holds
FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:33
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f, g being ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) & ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) [= g : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) holds
FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,g : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:34
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f, g being ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) & f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) holds
FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,g : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:35
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for v being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) holds
v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" (FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,( the L_join of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) [;] (v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

registration
let L be ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ;
cluster L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) .: : ( ( strict ) ( non empty strict ) LattStr ) -> strict Lattice-like ;
end;

theorem :: LATTICE2:36
for A being ( ( non empty ) ( non empty ) set )
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) )
for f9 being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = f9 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) holds
( FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f9 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f9 : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of (b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: LATTICE2:37
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for a9, b9 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = a9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = b9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = a9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" b9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = a9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" b9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: LATTICE2:38
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
for a9, b9 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = a9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = b9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
b9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= a9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:39
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for a9, b9 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= b9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = a9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = b9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:40
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for A being ( ( non empty ) ( non empty ) set )
for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:41
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st ex x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st
( x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) & f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) [= u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) holds
FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:42
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) & B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) holds
FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:43
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for v being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) holds
v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" (FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) = FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,( the L_meet of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) [;] (v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:44
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:45
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f, g being ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) & f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) : ( ( Function-like ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined b3 : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) holds
FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,g : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:46
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f being ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) & ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) holds
u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b3 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b3 : ( ( non empty ) ( non empty ) set ) ) V18(b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:47
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for f, g being ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) st B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) <> {} : ( ( ) ( Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty ) set ) & ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) [= g : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) holds
FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,g : ( ( Function-like V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b2 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b2 : ( ( non empty ) ( non empty ) set ) ) V18(b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b2 : ( ( non empty ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:48
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) holds
( L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) is lower-bounded iff L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .: : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) is upper-bounded ) ;

theorem :: LATTICE2:49
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) holds
( L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) is upper-bounded iff L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .: : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) is lower-bounded ) ;

theorem :: LATTICE2:50
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) holds
( L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) is ( ( non empty Lattice-like distributive ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular ) D_Lattice) iff L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .: : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) is ( ( non empty Lattice-like distributive ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular ) D_Lattice) ) ;

theorem :: LATTICE2:51
for L being ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) holds Bottom L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) is_a_unity_wrt the L_join of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;

registration
let L be ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) ;
cluster the L_join of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( Function-like V18(K20( the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) -> Function-like V18(K20( the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) having_a_unity ;
end;

theorem :: LATTICE2:52
for L being ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) holds Bottom L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) = the_unity_wrt the L_join of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent having_a_unity ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:53
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for L being ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice)
for f, g being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) holds
FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,g : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:54
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for L being ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice)
for f being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) )
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) [= u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) holds
FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:55
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for L being ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice)
for f, g being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) [= g : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) holds
FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,g : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:56
for L being ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) holds Top L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) is_a_unity_wrt the L_meet of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;

registration
let L be ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) ;
cluster the L_meet of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( Function-like V18(K20( the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) -> Function-like V18(K20( the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) LattStr ) : ( ( ) ( non empty ) set ) ) having_a_unity ;
end;

theorem :: LATTICE2:57
for L being ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) holds Top L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) = the_unity_wrt the L_meet of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent having_a_unity ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:58
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for L being ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice)
for f, g being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) | B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) : ( ( Function-like ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b2 : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) holds
FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,g : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:59
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for L being ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice)
for f being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) )
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) holds
u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:60
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for L being ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice)
for f, g being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) [= g : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) holds
FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= FinMeet (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,g : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:61
for L being ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) holds Bottom L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) = Top (L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( ) Element of the carrier of (b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:62
for L being ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) holds Top L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) : ( ( ) ( non empty ) set ) ) = Bottom (L : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( ) Element of the carrier of (b1 : ( ( non empty Lattice-like upper-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) 1_Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) ) ;

definition
mode D0_Lattice is ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) Lattice) ;
end;

theorem :: LATTICE2:63
for L being ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) holds the L_meet of L : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) is_distributive_wrt the L_join of L : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( Function-like V18(K20( the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent having_a_unity ) Element of K19(K20(K20( the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b1 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: LATTICE2:64
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for L being ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice)
for f being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) )
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds the L_meet of L : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( Function-like V18(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) . (u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,(FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) = FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,( the L_meet of L : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( Function-like V18(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) [;] (u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:65
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for L being ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice)
for g, f being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) )
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) Element of A : ( ( non empty ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) in B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) holds
g : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) = u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" (f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) holds
u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" (FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) = FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,g : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: LATTICE2:66
for A being ( ( non empty ) ( non empty ) set )
for B being ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) )
for L being ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice)
for f being ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of A : ( ( non empty ) ( non empty ) set ) , the carrier of L : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) )
for u being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" (FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) )) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) = FinJoin (B : ( ( ) ( ) Finite_Subset of ( ( preBoolean ) ( preBoolean ) set ) ) ,( the L_meet of L : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( Function-like V18(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V14(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) V18(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of K19(K20(K20( the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) [;] (u : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,f : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Function of b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) )) : ( ( Function-like V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like non empty V14(b1 : ( ( non empty ) ( non empty ) set ) ) V18(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) ) Element of K19(K20(b1 : ( ( non empty ) ( non empty ) set ) , the carrier of b3 : ( ( non empty Lattice-like distributive lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded ) D0_Lattice) : ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

definition
let IT be ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ;
attr IT is Heyting means :: LATTICE2:def 5
( IT : ( ( ) ( ) set ) is implicative & IT : ( ( ) ( ) set ) is lower-bounded );
end;

registration
cluster non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like Heyting for ( ( ) ( ) LattStr ) ;
end;

registration
cluster non empty Lattice-like Heyting -> non empty Lattice-like lower-bounded implicative for ( ( ) ( ) LattStr ) ;
cluster non empty Lattice-like lower-bounded implicative -> non empty Lattice-like Heyting for ( ( ) ( ) LattStr ) ;
end;

definition
mode H_Lattice is ( ( non empty Lattice-like Heyting ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded implicative Heyting ) Lattice) ;
end;

registration
cluster non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like Heyting for ( ( ) ( ) LattStr ) ;
end;

theorem :: LATTICE2:67
for L being ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) holds
( L : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) is ( ( non empty Lattice-like Heyting ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded implicative Heyting ) H_Lattice) iff for x, z being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex y being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) [= z : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & ( for v being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like lower-bounded ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded ) 0_Lattice) : ( ( ) ( non empty ) set ) ) [= z : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
v : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= y : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: LATTICE2:68
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) holds
( L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) is finite iff L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .: : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) is finite ) ;

registration
cluster non empty finite Lattice-like -> non empty Lattice-like lower-bounded for ( ( ) ( ) LattStr ) ;
cluster non empty finite Lattice-like -> non empty Lattice-like upper-bounded for ( ( ) ( ) LattStr ) ;
end;

registration
cluster non empty finite Lattice-like -> non empty Lattice-like bounded for ( ( ) ( ) LattStr ) ;
end;

registration
cluster non empty finite Lattice-like distributive -> non empty Lattice-like Heyting for ( ( ) ( ) LattStr ) ;
end;