:: FILTER_2 semantic presentation

begin

theorem :: FILTER_2:1
for D being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) )
for f being ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of D : ( ( non empty ) ( non empty ) set ) )
for g being ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) st g : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) = f : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) || S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) holds
( ( f : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) is commutative implies g : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) is commutative ) & ( f : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) is idempotent implies g : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) is idempotent ) & ( f : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) is associative implies g : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) is associative ) ) ;

theorem :: FILTER_2:2
for D being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) )
for f being ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of D : ( ( non empty ) ( non empty ) set ) )
for g being ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) )
for d being ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) )
for d9 being ( ( ) ( ) Element of S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) st g : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) = f : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) || S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) & d9 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) = d : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) holds
( ( d : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) is_a_left_unity_wrt f : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) implies d9 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) is_a_left_unity_wrt g : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) & ( d : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) is_a_right_unity_wrt f : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) implies d9 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) is_a_right_unity_wrt g : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) & ( d : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) is_a_unity_wrt f : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) implies d9 : ( ( ) ( ) Element of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) is_a_unity_wrt g : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: FILTER_2:3
for D being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) )
for f1, f2 being ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of D : ( ( non empty ) ( non empty ) set ) )
for g1, g2 being ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) st g1 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) = f1 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) || S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) & g2 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) = f2 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) || S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) holds
( ( f1 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) is_left_distributive_wrt f2 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) implies g1 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) is_left_distributive_wrt g2 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) & ( f1 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) is_right_distributive_wrt f2 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) implies g1 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) is_right_distributive_wrt g2 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: FILTER_2:4
for D being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) )
for f1, f2 being ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of D : ( ( non empty ) ( non empty ) set ) )
for g1, g2 being ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) st g1 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) = f1 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) || S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) & g2 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) = f2 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) || S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) & f1 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) is_distributive_wrt f2 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) holds
g1 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) is_distributive_wrt g2 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ;

theorem :: FILTER_2:5
for D being ( ( non empty ) ( non empty ) set )
for S being ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) )
for f1, f2 being ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of D : ( ( non empty ) ( non empty ) set ) )
for g1, g2 being ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) st g1 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) = f1 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) || S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) & g2 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) = f2 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) || S : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) set ) & f1 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) absorbs f2 : ( ( Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like [:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like V18([:b1 : ( ( non empty ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) BinOp of b1 : ( ( non empty ) ( non empty ) set ) ) holds
g1 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) absorbs g2 : ( ( Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) ( Relation-like [:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -valued Function-like V18([:b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ) BinOp of b2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) ;

begin

definition
let D be ( ( non empty ) ( non empty ) set ) ;
let X1, X2 be ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ;
:: original: =
redefine pred X1 = X2 means :: FILTER_2:def 1
 for x being ( ( ) ( ) Element of D : ( ( ) ( ) LattStr ) ) holds
( x : ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) in X1 : ( ( Function-like V18([:D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,D : ( ( ) ( ) LattStr ) ) ) ( Relation-like [:D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) -defined D : ( ( ) ( ) LattStr ) -valued Function-like V18([:D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,D : ( ( ) ( ) LattStr ) ) ) Element of bool [:[:D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,D : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) iff x : ( ( ) ( ) Element of D : ( ( non empty ) ( non empty ) set ) ) in X2 : ( ( Function-like V18([:D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,D : ( ( ) ( ) LattStr ) ) ) ( Relation-like [:D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) -defined D : ( ( ) ( ) LattStr ) -valued Function-like V18([:D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,D : ( ( ) ( ) LattStr ) ) ) Element of bool [:[:D : ( ( ) ( ) LattStr ) ,D : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,D : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) );
end;

theorem :: FILTER_2:6
for L1, L2 being ( ( ) ( ) LattStr ) st LattStr(# the carrier of L1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the L_join of L1 : ( ( ) ( ) LattStr ) : ( ( Function-like V18([: the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) -valued Function-like V18([: the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L1 : ( ( ) ( ) LattStr ) : ( ( Function-like V18([: the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) -valued Function-like V18([: the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) = LattStr(# the carrier of L2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the L_join of L2 : ( ( ) ( ) LattStr ) : ( ( Function-like V18([: the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) ( Relation-like [: the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) -valued Function-like V18([: the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L2 : ( ( ) ( ) LattStr ) : ( ( Function-like V18([: the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) ( Relation-like [: the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) -valued Function-like V18([: the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( ) ( ) LattStr ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) holds
L1 : ( ( ) ( ) LattStr ) .: : ( ( strict ) ( strict ) LattStr ) = L2 : ( ( ) ( ) LattStr ) .: : ( ( strict ) ( strict ) LattStr ) ;

theorem :: FILTER_2:7
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) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) .: : ( ( strict ) ( strict ) LattStr ) = LattStr(# 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 ) , 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) ;

theorem :: FILTER_2:8
for L1, L2 being ( ( non empty ) ( non empty ) LattStr ) st LattStr(# the carrier of L1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the L_join of L1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) = LattStr(# the carrier of L2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the L_join of L2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18([: the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( Function-like V18([: the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) ) Element of bool [:[: the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) holds
for a1, b1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for a2, b2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = a2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = b2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( a1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" b1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) = a2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" b2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) & a1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" b1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) = a2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" b2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty ) ( non empty ) LattStr ) : ( ( ) ( non empty ) set ) ) & ( a1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= b1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) implies a2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= b2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) & ( a2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= b2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) implies a1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= b1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: FILTER_2:9
for L1, L2 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) st LattStr(# the carrier of L1 : ( ( 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 L_join of L1 : ( ( 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([: 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 ) :] : ( ( ) ( ) 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 [: 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 ) :] : ( ( ) ( ) 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 V18([: 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 ) :] : ( ( ) ( ) 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 bool [:[: 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 ) :] : ( ( ) ( ) 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L1 : ( ( 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([: 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 ) :] : ( ( ) ( ) 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 [: 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 ) :] : ( ( ) ( ) 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 V18([: 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 ) :] : ( ( ) ( ) 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 bool [:[: 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 ) :] : ( ( ) ( ) 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) = LattStr(# the carrier of L2 : ( ( 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 L_join of L2 : ( ( 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([: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 [: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( 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 V18([: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 bool [:[: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L2 : ( ( 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([: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 [: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( 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 V18([: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 bool [:[: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) holds
Bottom L1 : ( ( 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 ) ) = Bottom L2 : ( ( 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 b2 : ( ( 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 :: FILTER_2:10
for L1, L2 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) st LattStr(# the carrier of L1 : ( ( 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 L_join of L1 : ( ( 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([: 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 ) :] : ( ( ) ( ) 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 [: 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 ) :] : ( ( ) ( ) 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 V18([: 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 ) :] : ( ( ) ( ) 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 bool [:[: 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 ) :] : ( ( ) ( ) 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L1 : ( ( 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([: 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 ) :] : ( ( ) ( ) 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 [: 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 ) :] : ( ( ) ( ) 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 V18([: 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 ) :] : ( ( ) ( ) 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 bool [:[: 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 ) :] : ( ( ) ( ) 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) = LattStr(# the carrier of L2 : ( ( 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 L_join of L2 : ( ( 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([: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 [: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( 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 V18([: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 bool [:[: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L2 : ( ( 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([: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 [: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( 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 V18([: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 bool [:[: the carrier of b2 : ( ( 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 b2 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) holds
Top L1 : ( ( 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 ) ) = Top L2 : ( ( 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 b2 : ( ( 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 :: FILTER_2:11
for L1, L2 being ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) st LattStr(# the carrier of L1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the L_join of L1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of bool [:[: the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of bool [:[: the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) = LattStr(# the carrier of L2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the L_join of L2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( Function-like V18([: the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of bool [:[: the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( Function-like V18([: the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of bool [:[: the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) holds
for a1, b1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for a2, b2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = a2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & b1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = b2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & a1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_a_complement_of b1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
a2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is_a_complement_of b2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: FILTER_2:12
for L1, L2 being ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) st LattStr(# the carrier of L1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the L_join of L1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of bool [:[: the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( Function-like V18([: the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of bool [:[: the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) = LattStr(# the carrier of L2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the L_join of L2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( Function-like V18([: the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of bool [:[: the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( Function-like V18([: the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) ) ( Relation-like [: the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) -defined the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) -valued Function-like V18([: the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) commutative associative idempotent ) Element of bool [:[: the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) , the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) holds
for a being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for b being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ` : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) = b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ` : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) ;

theorem :: FILTER_2:13
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 X being ( ( ) ( ) Subset of ) st ( for p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( ( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Subset of ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Subset of ) ) iff p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" q : ( ( ) ( ) 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 ) ) in X : ( ( ) ( ) Subset of ) ) ) holds
X : ( ( ) ( ) Subset of ) is ( ( meet-closed join-closed ) ( meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:14
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 X being ( ( ) ( ) Subset of ) st ( for p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( ( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Subset of ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in X : ( ( ) ( ) Subset of ) ) iff p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" q : ( ( ) ( ) 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 ) ) in X : ( ( ) ( ) Subset of ) ) ) holds
X : ( ( ) ( ) Subset of ) is ( ( meet-closed join-closed ) ( meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

definition
let L be ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ;
mode Ideal of L is ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Subset of ) ;
end;

theorem :: FILTER_2:15
for L1, L2 being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) st LattStr(# the carrier of L1 : ( ( 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 L_join of L1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) = LattStr(# the carrier of L2 : ( ( 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 L_join of L2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: 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 ) , 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 ) :] : ( ( ) ( ) 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 [: 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 ) , 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 ) :] : ( ( ) ( ) 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 V18([: 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 ) , 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 ) :] : ( ( ) ( ) 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 ) ) commutative associative idempotent ) Element of bool [:[: 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 ) , 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 ) :] : ( ( ) ( ) 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: 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 ) , 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 ) :] : ( ( ) ( ) 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 [: 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 ) , 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 ) :] : ( ( ) ( ) 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 V18([: 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 ) , 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 ) :] : ( ( ) ( ) 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 ) ) commutative associative idempotent ) Element of bool [:[: 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 ) , 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 ) :] : ( ( ) ( ) 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) holds
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) is ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter of L1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds
x : ( ( ) ( ) set ) is ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter of L2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:16
for L1, L2 being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) st LattStr(# the carrier of L1 : ( ( 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 L_join of L1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) = LattStr(# the carrier of L2 : ( ( 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 L_join of L2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: 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 ) , 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 ) :] : ( ( ) ( ) 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 [: 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 ) , 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 ) :] : ( ( ) ( ) 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 V18([: 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 ) , 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 ) :] : ( ( ) ( ) 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 ) ) commutative associative idempotent ) Element of bool [:[: 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 ) , 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 ) :] : ( ( ) ( ) 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: 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 ) , 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 ) :] : ( ( ) ( ) 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 [: 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 ) , 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 ) :] : ( ( ) ( ) 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 V18([: 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 ) , 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 ) :] : ( ( ) ( ) 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 ) ) commutative associative idempotent ) Element of bool [:[: 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 ) , 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 ) :] : ( ( ) ( ) 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 ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) holds
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) is ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds
x : ( ( ) ( ) set ) is ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

definition
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 p be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
func p .: -> ( ( ) ( ) Element of ( ( ) ( ) set ) ) equals :: FILTER_2:def 2
p : ( ( Function-like V18([:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) ) ) ( Relation-like [:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) -defined L : ( ( ) ( ) LattStr ) -valued Function-like V18([:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) ) ) Element of bool [:[:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
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 p be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
func .: p -> ( ( ) ( ) Element of ( ( ) ( ) set ) ) equals :: FILTER_2:def 3
p : ( ( Function-like V18([:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) ) ) ( Relation-like [:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) -defined L : ( ( ) ( ) LattStr ) -valued Function-like V18([:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) ) ) Element of bool [:[:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: FILTER_2:17
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for p9 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
( .: (p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .:) : ( ( ) ( ) Element of ( ( ) ( ) set ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & (.: p9 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .: : ( ( ) ( ) Element of ( ( ) ( ) set ) ) = p9 : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) ;

theorem :: FILTER_2:18
errorfrm ;

theorem :: FILTER_2:19
errorfrm ;

definition
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 X be ( ( ) ( ) Subset of ) ;
func X .: -> ( ( ) ( ) Subset of ) equals :: FILTER_2:def 4
X : ( ( Function-like V18([:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) ) ) ( Relation-like [:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) -defined L : ( ( ) ( ) LattStr ) -valued Function-like V18([:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) ) ) Element of bool [:[:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
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 X be ( ( ) ( ) Subset of ) ;
func .: X -> ( ( ) ( ) Subset of ) equals :: FILTER_2:def 5
X : ( ( Function-like V18([:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) ) ) ( Relation-like [:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) -defined L : ( ( ) ( ) LattStr ) -valued Function-like V18([:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) ) ) Element of bool [:[:L : ( ( ) ( ) LattStr ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( ) ( ) LattStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ;
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) ;
let D be ( ( non empty ) ( non empty ) Subset of ) ;
cluster D : ( ( non empty ) ( non empty ) Element of bool 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 ) : ( ( ) ( non empty ) set ) ) .: : ( ( ) ( ) Subset of ) -> non empty ;
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) ;
let D be ( ( non empty ) ( non empty ) Subset of ) ;
cluster .: D : ( ( non empty ) ( non empty ) Element of bool 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 ) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Subset of ) -> non empty ;
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) ;
let S be ( ( meet-closed ) ( meet-closed ) Subset of ) ;
cluster S : ( ( meet-closed ) ( meet-closed ) Element of bool 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 ) : ( ( ) ( non empty ) set ) ) .: : ( ( ) ( ) Subset of ) -> join-closed for ( ( ) ( ) Subset of ) ;
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) ;
let S be ( ( join-closed ) ( join-closed ) Subset of ) ;
cluster S : ( ( join-closed ) ( join-closed ) Element of bool 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 ) : ( ( ) ( non empty ) set ) ) .: : ( ( ) ( ) Subset of ) -> meet-closed for ( ( ) ( ) Subset of ) ;
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) ;
let S be ( ( meet-closed ) ( meet-closed ) Subset of ) ;
cluster .: S : ( ( meet-closed ) ( meet-closed ) Element of bool 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 ) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Subset of ) -> join-closed for ( ( ) ( ) Subset of ) ;
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) ;
let S be ( ( join-closed ) ( join-closed ) Subset of ) ;
cluster .: S : ( ( join-closed ) ( join-closed ) Element of bool 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 ) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Subset of ) -> meet-closed for ( ( ) ( ) Subset of ) ;
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) ;
let F be ( ( final ) ( final join-closed ) Subset of ) ;
cluster F : ( ( final ) ( final join-closed ) Element of bool 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 ) : ( ( ) ( non empty ) set ) ) .: : ( ( ) ( meet-closed ) Subset of ) -> initial for ( ( ) ( ) Subset of ) ;
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) ;
let F be ( ( initial ) ( initial meet-closed ) Subset of ) ;
cluster F : ( ( initial ) ( initial meet-closed ) Element of bool 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 ) : ( ( ) ( non empty ) set ) ) .: : ( ( ) ( join-closed ) Subset of ) -> final for ( ( ) ( ) Subset of ) ;
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) ;
let F be ( ( final ) ( final join-closed ) Subset of ) ;
cluster .: F : ( ( final ) ( final join-closed ) Element of bool 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 ) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( meet-closed ) Subset of ) -> initial for ( ( ) ( ) Subset of ) ;
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) ;
let F be ( ( initial ) ( initial meet-closed ) Subset of ) ;
cluster F : ( ( initial ) ( initial meet-closed ) Element of bool 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 ) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) .: : ( ( ) ( final join-closed ) Subset of ) -> final for ( ( ) ( ) Subset of ) ;
end;

theorem :: FILTER_2:20
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for x being ( ( ) ( ) set ) holds
( x : ( ( ) ( ) set ) is ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) iff x : ( ( ) ( ) set ) is ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter 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 ) ) ) ;

theorem :: FILTER_2:21
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for D being ( ( non empty ) ( non empty ) Subset of ) holds
( D : ( ( non empty ) ( non empty ) Subset of ) is ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) iff ( ( for p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in D : ( ( non empty ) ( non empty ) Subset of ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in D : ( ( non empty ) ( non empty ) Subset of ) holds
p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" q : ( ( ) ( ) 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 ) ) in D : ( ( non empty ) ( non empty ) Subset of ) ) & ( for p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in D : ( ( non empty ) ( non empty ) Subset of ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in D : ( ( non empty ) ( non empty ) Subset of ) ) ) ) ;

theorem :: FILTER_2:22
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for I being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" q : ( ( ) ( ) 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 ) ) in I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" p : ( ( ) ( ) 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 ) ) in I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ;

theorem :: FILTER_2:23
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for I being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ex p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:24
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for I being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( 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 lower-bounded 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 ) ) in I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:25
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) st 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 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 ) ) } : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:26
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st {p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) is ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( 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 ;

begin

theorem :: FILTER_2:27
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 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 ) is ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

definition
let L be ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ;
func (.L.> -> ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) equals :: FILTER_2:def 6
 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 ) ;
end;

definition
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 p be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
func (.p.> -> ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) equals :: FILTER_2:def 7
 { q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) where q is ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= p : ( ( ) ( ) set ) } ;
end;

theorem :: FILTER_2:28
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 q, p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in (.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) iff q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: FILTER_2:29
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( (.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = <.(p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .:) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .) : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Element of bool the 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 ) : ( ( ) ( non empty ) set ) ) & (.(p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .:) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal 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 ) ) = <.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .) : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: FILTER_2:30
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 p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in (.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" q : ( ( ) ( ) 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 ) ) in (.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" p : ( ( ) ( ) 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 ) ) in (.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ;

theorem :: FILTER_2:31
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) st 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 holds
(.L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = (.(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 ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

definition
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 I be ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;
pred I is_max-ideal means :: FILTER_2:def 8
( I : ( ( ) ( ) 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 ) & ( for J being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) st I : ( ( ) ( ) set ) c= J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) <> 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 ) holds
I : ( ( ) ( ) set ) = J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) );
end;

theorem :: FILTER_2:32
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 I being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds
( I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) is_max-ideal iff I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) .: : ( ( ) ( non empty final meet-closed join-closed ) Subset of ) is being_ultrafilter ) ;

theorem :: FILTER_2:33
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) st 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 holds
for I being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) st I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) <> 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 ) holds
ex J being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) st
( I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) c= J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) is_max-ideal ) ;

theorem :: FILTER_2:34
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st ex r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" r : ( ( ) ( ) 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 ) ) <> p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
(.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) <> 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 ) ;

theorem :: FILTER_2:35
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st 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 & p : ( ( ) ( ) 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 ) ) holds
ex I being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) st
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) is_max-ideal ) ;

definition
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 D be ( ( non empty ) ( non empty ) Subset of ) ;
func (.D.> -> ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) means :: FILTER_2:def 9
( D : ( ( ) ( ) set ) c= it : ( ( Function-like V18([:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) ) ( Relation-like [:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) -defined L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -valued Function-like V18([:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) ) Element of bool [:[:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) & ( for I being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) st D : ( ( ) ( ) set ) c= I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds
it : ( ( Function-like V18([:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) ) ( Relation-like [:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) -defined L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -valued Function-like V18([:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) ) Element of bool [:[:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) c= I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) );
end;

theorem :: FILTER_2:36
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 D being ( ( non empty ) ( non empty ) Subset of )
for D9 being ( ( non empty ) ( non empty ) Subset of ) holds
( <.(D : ( ( non empty ) ( non empty ) Subset of ) .:) : ( ( ) ( non empty ) Subset of ) .) : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Element of bool the 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 ) : ( ( ) ( non empty ) set ) ) = (.D : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & <.D : ( ( non empty ) ( non empty ) Subset of ) .) : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (.(D : ( ( non empty ) ( non empty ) Subset of ) .:) : ( ( ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal 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 ) ) & <.(.: D9 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( non empty ) Subset of ) .) : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (.D9 : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal 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 ) ) & <.D9 : ( ( non empty ) ( non empty ) Subset of ) .) : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Element of bool the 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 ) : ( ( ) ( non empty ) set ) ) = (.(.: D9 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ;

theorem :: FILTER_2:37
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 I being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds (.I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:38
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 D, D1, D2 being ( ( non empty ) ( non empty ) Subset of ) holds
( ( D1 : ( ( non empty ) ( non empty ) Subset of ) c= D2 : ( ( non empty ) ( non empty ) Subset of ) implies (.D1 : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) c= (.D2 : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) & (.(.D : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) c= (.D : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ;

theorem :: FILTER_2:39
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in D : ( ( non empty ) ( non empty ) Subset of ) holds
(.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) c= (.D : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:40
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for D being ( ( non empty ) ( non empty ) Subset of ) st D : ( ( non empty ) ( non empty ) Subset of ) = {p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
(.D : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = (.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:41
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 D being ( ( non empty ) ( non empty ) Subset of ) st 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 & 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 ) ) in D : ( ( non empty ) ( non empty ) Subset of ) holds
( (.D : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = (.L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & (.D : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = 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 ) ) ;

theorem :: FILTER_2:42
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 I being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( 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 upper-bounded & 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 ) ) in I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds
( I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = (.L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = 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 ) ) ;

definition
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 I be ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;
attr I is prime means :: FILTER_2:def 10
 for p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of 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 ) ) in I : ( ( ) ( ) set ) iff ( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in I : ( ( ) ( ) set ) or q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in I : ( ( ) ( ) set ) ) );
end;

theorem :: FILTER_2:43
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 I being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds
( I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) is prime iff I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) .: : ( ( ) ( non empty final meet-closed join-closed ) Subset of ) is prime ) ;

definition
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 D1, D2 be ( ( non empty ) ( non empty ) Subset of ) ;
func D1 "\/" D2 -> ( ( ) ( ) Subset of ) equals :: FILTER_2:def 11
 { (p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of 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 ) ) where p, q is ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in D1 : ( ( ) ( ) set ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in D2 : ( ( Function-like V18([:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) ) ( Relation-like [:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) -defined L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -valued Function-like V18([:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) ) Element of bool [:[:L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) ,L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) ) } ;
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) ;
let D1, D2 be ( ( non empty ) ( non empty ) Subset of ) ;
cluster D1 : ( ( non empty ) ( non empty ) Element of bool 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 ) : ( ( ) ( non empty ) set ) ) "\/" D2 : ( ( non empty ) ( non empty ) Element of bool 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 ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Subset of ) -> non empty ;
end;

theorem :: FILTER_2:44
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 D1, D2 being ( ( non empty ) ( non empty ) Subset of )
for D19, D29 being ( ( non empty ) ( non empty ) Subset of ) holds
( D1 : ( ( non empty ) ( non empty ) Subset of ) "\/" D2 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Subset of ) = (D1 : ( ( non empty ) ( non empty ) Subset of ) .:) : ( ( ) ( non empty ) Subset of ) "/\" (D2 : ( ( non empty ) ( non empty ) Subset of ) .:) : ( ( ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Element of bool the 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 ) : ( ( ) ( non empty ) set ) ) & (D1 : ( ( non empty ) ( non empty ) Subset of ) .:) : ( ( ) ( non empty ) Subset of ) "\/" (D2 : ( ( non empty ) ( non empty ) Subset of ) .:) : ( ( ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Subset of ) = D1 : ( ( non empty ) ( non empty ) Subset of ) "/\" D2 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & D19 : ( ( non empty ) ( non empty ) Subset of ) "\/" D29 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Subset of ) = (.: D19 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( non empty ) Subset of ) "/\" (.: D29 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & (.: D19 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( non empty ) Subset of ) "\/" (.: D29 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Subset of ) = D19 : ( ( non empty ) ( non empty ) Subset of ) "/\" D29 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Element of bool the 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 ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: FILTER_2:45
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 p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for D1, D2 being ( ( non empty ) ( non empty ) Subset of ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in D1 : ( ( non empty ) ( non empty ) Subset of ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in D2 : ( ( non empty ) ( non empty ) Subset of ) holds
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" q : ( ( ) ( ) 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 ) ) in D1 : ( ( non empty ) ( non empty ) Subset of ) "\/" D2 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Subset of ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" p : ( ( ) ( ) 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 ) ) in D1 : ( ( non empty ) ( non empty ) Subset of ) "\/" D2 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Subset of ) ) ;

theorem :: FILTER_2:46
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 x being ( ( ) ( ) set )
for D1, D2 being ( ( non empty ) ( non empty ) Subset of ) st x : ( ( ) ( ) set ) in D1 : ( ( non empty ) ( non empty ) Subset of ) "\/" D2 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Subset of ) holds
ex p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( x : ( ( ) ( ) set ) = p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" q : ( ( ) ( ) 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 ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in D1 : ( ( non empty ) ( non empty ) Subset of ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in D2 : ( ( non empty ) ( non empty ) Subset of ) ) ;

theorem :: FILTER_2:47
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 D1, D2 being ( ( non empty ) ( non empty ) Subset of ) holds D1 : ( ( non empty ) ( non empty ) Subset of ) "\/" D2 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Subset of ) = D2 : ( ( non empty ) ( non empty ) Subset of ) "\/" D1 : ( ( non empty ) ( non empty ) Subset of ) : ( ( ) ( non empty ) Subset of ) ;

registration
let L be ( ( 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) ;
let I1, I2 be ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( 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) ) ;
cluster I1 : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Element of bool the carrier of L : ( ( non empty Lattice-like distributive ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular ) LattStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) "\/" I2 : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Element of bool the carrier of L : ( ( non empty Lattice-like distributive ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular ) LattStr ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Subset of ) -> initial join-closed ;
end;

theorem :: FILTER_2:48
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for D1, D2 being ( ( non empty ) ( non empty ) Subset of ) holds
( (.(D1 : ( ( non empty ) ( non empty ) Subset of ) \/ D2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = (.((.D1 : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) \/ D2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & (.(D1 : ( ( non empty ) ( non empty ) Subset of ) \/ D2 : ( ( non empty ) ( non empty ) Subset of ) ) : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = (.(D1 : ( ( non empty ) ( non empty ) Subset of ) \/ (.D2 : ( ( non empty ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ;

theorem :: FILTER_2:49
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for I, J being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds (.(I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) \/ J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = { r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) where r is ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ex p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st
( r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" q : ( ( ) ( ) 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 ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) } ;

theorem :: FILTER_2:50
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for I, J being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds
( I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) c= I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) "\/" J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) Subset of ) & J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) c= I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) "\/" J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) Subset of ) ) ;

theorem :: FILTER_2:51
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 I, J being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds (.(I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) \/ J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = (.(I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) "\/" J : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty ) Subset of ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:52
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 bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_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 bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) ) ;

theorem :: FILTER_2:53
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 Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_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 Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) ;

registration
let B be ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ;
cluster B : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) LattStr ) .: : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -> strict Lattice-like Boolean ;
end;

theorem :: FILTER_2:54
for B being ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice)
for a being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for a9 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( (a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .:) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ` : ( ( ) ( ) Element of the carrier of (b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) = a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ` : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) & (.: a9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ` : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) = a9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ` : ( ( ) ( ) Element of the carrier of (b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) .:) : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) LattStr ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: FILTER_2:55
for B being ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice)
for IB, JB being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of B : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) holds (.(IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) \/ JB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) ) : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) = IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) "\/" JB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) : ( ( ) ( non empty initial meet-closed join-closed ) Subset of ) ;

theorem :: FILTER_2:56
for B being ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice)
for IB being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of B : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) holds
( IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) is_max-ideal iff ( IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) <> the carrier of B : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) & ( for a being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) or a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ` : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) in IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) ) ) ) ) ;

theorem :: FILTER_2:57
for B being ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice)
for IB being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of B : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) holds
( ( IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) <> (.B : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) & IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) is prime ) iff IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) is_max-ideal ) ;

theorem :: FILTER_2:58
for B being ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice)
for IB being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of B : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) st IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) is_max-ideal holds
for a being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) iff not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ` : ( ( ) ( ) Element of the carrier of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) : ( ( ) ( non empty ) set ) ) in IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) ) ;

theorem :: FILTER_2:59
for B being ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice)
for a, b being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) <> b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
ex IB being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of B : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) st
( IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) is_max-ideal & ( ( a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) & not b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) ) or ( not a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) & b : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in IB : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ) ) ) ) ;

theorem :: FILTER_2:60
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 P being ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( Relation-like Function-like ) set ) is ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) & 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( Relation-like Function-like ) set ) is ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ;

theorem :: FILTER_2:61
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 P being ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) )
for o1, o2 being ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) st o1 : ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) = 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( Relation-like Function-like ) set ) & o2 : ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) = 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( Relation-like Function-like ) set ) holds
( o1 : ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) is commutative & o1 : ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) is associative & o2 : ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) is commutative & o2 : ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) is associative & o1 : ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) absorbs o2 : ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) & o2 : ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) absorbs o1 : ( ( Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) -defined b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) :] : ( ( ) ( non empty ) set ) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ;

definition
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 p, q be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
assume p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
func [#p,q#] -> ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) equals :: FILTER_2:def 12
 { r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) where r is ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( p : ( ( ) ( ) set ) [= r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= q : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Element of bool 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 ) : ( ( ) ( non empty ) set ) ) ) } ;
end;

theorem :: FILTER_2:62
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 p, q, r being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in [#p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) iff ( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & r : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: FILTER_2:63
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 p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in [#p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in [#p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ;

theorem :: FILTER_2:64
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds [#p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = {p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) } : ( ( ) ( non empty ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: FILTER_2:65
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st 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 holds
<.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .) : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = [#p : ( ( ) ( ) 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 ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:66
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st 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 holds
(.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = [#(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 ) ) ,p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:67
for L1, L2 being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for F1 being ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter of L1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) )
for F2 being ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter of L2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) st LattStr(# the carrier of L1 : ( ( 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 L_join of L1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) LattStr ) = LattStr(# the carrier of L2 : ( ( 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 L_join of L2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: 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 ) , 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 ) :] : ( ( ) ( 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 [: 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 ) , 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 ) :] : ( ( ) ( 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 V18([: 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 ) , 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 ) :] : ( ( ) ( 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 ) ) commutative associative idempotent ) Element of bool [:[: 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 ) , 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 ) :] : ( ( ) ( 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 ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of L2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( Function-like V18([: 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 ) , 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 ) :] : ( ( ) ( 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 [: 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 ) , 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 ) :] : ( ( ) ( 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 V18([: 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 ) , 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 ) :] : ( ( ) ( 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 ) ) commutative associative idempotent ) Element of bool [:[: 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 ) , 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 ) :] : ( ( ) ( 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 ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) LattStr ) & F1 : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = F2 : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter of b2 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds
latt F1 : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( non empty Lattice-like ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) = latt F2 : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter 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 Lattice-like ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ;

begin

notation
let L be ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ;
synonym Sublattice of L for SubLattice of L;
end;

definition
let L be ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ;
redefine mode SubLattice of L means :: FILTER_2:def 13
 ex P being ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) ex o1, o2 being ( ( Function-like V18([:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) -defined b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) st
( o1 : ( ( Function-like V18([:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) -defined b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) = 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([: 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 ) :] : ( ( ) ( 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 [: 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 ) :] : ( ( ) ( 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 ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V18([: 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 ) :] : ( ( ) ( 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 ) ) commutative associative idempotent ) Element of bool [:[: 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 ) :] : ( ( ) ( 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 ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( Relation-like Function-like ) set ) & o2 : ( ( Function-like V18([:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) -defined b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) = 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([: 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 ) :] : ( ( ) ( 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 [: 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 ) :] : ( ( ) ( 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 ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V18([: 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 ) :] : ( ( ) ( 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 ) ) commutative associative idempotent ) Element of bool [:[: 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 ) :] : ( ( ) ( 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 ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( Relation-like Function-like ) set ) & LattStr(# the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the L_join of it : ( ( ) ( ) set ) : ( ( Function-like V18([: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like [: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like V18([: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) , the L_meet of it : ( ( ) ( ) set ) : ( ( Function-like V18([: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like [: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like V18([: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) ) ) Element of bool [:[: the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) , the carrier of it : ( ( ) ( ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) LattStr ) = LattStr(# P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,o1 : ( ( Function-like V18([:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) -defined b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ,o2 : ( ( Function-like V18([:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) -defined b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of b1 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) LattStr ) );
end;

theorem :: FILTER_2:68
for L being ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice)
for K being ( ( ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) )
for a being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds a : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

definition
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 P be ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;
func latt (L,P) -> ( ( ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) means :: FILTER_2:def 14
 ex o1, o2 being ( ( Function-like V18([:P : ( ( ) ( ) set ) ,P : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,P : ( ( ) ( ) set ) ) ) ( Relation-like [:P : ( ( ) ( ) set ) ,P : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) -defined P : ( ( ) ( ) set ) -valued Function-like V18([:P : ( ( ) ( ) set ) ,P : ( ( ) ( ) set ) :] : ( ( ) ( ) set ) ,P : ( ( ) ( ) set ) ) ) BinOp of P : ( ( ) ( ) set ) ) st
( o1 : ( ( Function-like V18([:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) -defined P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) = 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([: 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 ) :] : ( ( ) ( 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 [: 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 ) :] : ( ( ) ( 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 ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V18([: 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 ) :] : ( ( ) ( 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 ) ) commutative associative idempotent ) Element of bool [:[: 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 ) :] : ( ( ) ( 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 ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || P : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like ) set ) & o2 : ( ( Function-like V18([:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) -defined P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) = 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([: 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 ) :] : ( ( ) ( 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 [: 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 ) :] : ( ( ) ( 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 ) LattStr ) : ( ( ) ( non empty ) set ) -valued Function-like V18([: 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 ) :] : ( ( ) ( 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 ) ) commutative associative idempotent ) Element of bool [:[: 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 ) :] : ( ( ) ( 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 ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || P : ( ( ) ( ) set ) : ( ( ) ( Relation-like Function-like ) set ) & it : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Element of bool 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 ) : ( ( ) ( non empty ) set ) ) = LattStr(# P : ( ( ) ( ) set ) ,o1 : ( ( Function-like V18([:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) -defined P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ,o2 : ( ( Function-like V18([:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) ( Relation-like [:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) -defined P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) -valued Function-like V18([:P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset 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 ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) ) BinOp of P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) #) : ( ( strict ) ( strict ) LattStr ) );
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) ;
let P be ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;
cluster latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) Element of bool 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 ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) -> strict ;
end;

definition
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 l be ( ( ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;
:: original: .:
redefine func l .: -> ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of 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 join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) ;
end;

theorem :: FILTER_2:69
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 F being ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds latt F : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( non empty Lattice-like ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) = latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,F : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Filter of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ;

theorem :: FILTER_2:70
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 P being ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = (latt ((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 ) ,(P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) .:) : ( ( ) ( non empty meet-closed join-closed ) Subset of ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice 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 ) ) .: : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice 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 ) .: : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) ;

theorem :: FILTER_2:71
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
( latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,(.L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = LattStr(# 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 ) , 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) LattStr ) & latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,<.L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) .) : ( ( non empty final meet-closed ) ( non empty final meet-closed join-closed ) Element of bool the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) = LattStr(# 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 ) , 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict ) LattStr ) ) ;

theorem :: FILTER_2:72
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 P being ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) holds
( the carrier of (latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) = P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) & the L_join of (latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( Function-like V18([: the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 [: the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 V18([: the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 bool [:[: the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( Relation-like Function-like ) set ) & the L_meet of (latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( Function-like V18([: the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 [: the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) -defined the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 V18([: the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 bool [:[: the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) , the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) 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([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 [: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( 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 V18([: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( 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 bool [:[: the carrier of b1 : ( ( 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 ) :] : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) || P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( Relation-like Function-like ) set ) ) ;

theorem :: FILTER_2:73
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 P being ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) )
for p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for p9, q9 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = p9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = q9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" q : ( ( ) ( ) 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 ) ) = p9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" q9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) ) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" q : ( ( ) ( ) 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 ) ) = p9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "/\" q9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,b2 : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 :: FILTER_2:74
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 P being ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) )
for p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for p9, q9 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = p9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) = q9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) iff p9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= q9 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: FILTER_2:75
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 I being ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( 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 lower-bounded holds
latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,I : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) is lower-bounded ;

theorem :: FILTER_2:76
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 P being ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( 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 modular holds
latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) is modular ;

theorem :: FILTER_2:77
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 P being ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of L : ( ( 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 distributive holds
latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,P : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) is distributive ;

theorem :: FILTER_2:78
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 p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) is implicative & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,[#p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) is implicative ;

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) ;
let p be ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;
cluster latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,(.p : ( ( ) ( ) Element of 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 ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal 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 strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) -> upper-bounded ;
end;

theorem :: FILTER_2:79
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds Top (latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,(.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,(.b2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) ) = p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ;

theorem :: FILTER_2:80
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st 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 holds
( latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,(.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) is lower-bounded & Bottom (latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,(.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,(.b2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( 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 :: FILTER_2:81
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st 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 holds
latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,(.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) is bounded ;

theorem :: FILTER_2:82
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 p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,[#p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) is bounded & Top (latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,[#p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,[#b2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) ) = q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) & Bottom (latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,[#p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 (latt (b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,[#b2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,b3 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) )) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) : ( ( ) ( non empty ) set ) ) = p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) ;

theorem :: FILTER_2:83
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 p being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) 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 bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) & L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) is modular holds
latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,(.p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) .> : ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded ) Sublattice of b1 : ( ( 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 bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) ;

theorem :: FILTER_2:84
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 p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) 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 bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) & L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) is modular & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,[#p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 bounded complemented ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented ) C_Lattice) ;

theorem :: FILTER_2:85
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 p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) 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 Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) & p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) [= q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
latt (L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ,[#p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) #] : ( ( non empty meet-closed join-closed ) ( non empty meet-closed join-closed ) ClosedSubset of b1 : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) ) : ( ( ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Sublattice of b1 : ( ( 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 Boolean ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting ) B_Lattice) ;

theorem :: FILTER_2:86
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 S being ( ( non empty ) ( non empty ) Subset of ) holds
( S : ( ( non empty ) ( non empty ) Subset of ) is ( ( non empty initial join-closed ) ( non empty initial meet-closed join-closed ) Ideal of L : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) Lattice) ) iff for p, q being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds
( ( p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in S : ( ( non empty ) ( non empty ) Subset of ) & q : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) in S : ( ( non empty ) ( non empty ) Subset of ) ) iff p : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) "\/" q : ( ( ) ( ) 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 ) ) in S : ( ( non empty ) ( non empty ) Subset of ) ) ) ;