let L be Lattice; :: thesis: for u being Element of L
for A being non empty set
for B being Finite_Subset of A
for f being Function of A,the carrier of L st B <> {} & ( for x being Element of A st x in B holds
f . x [= u ) holds
FinJoin B,f [= u
let u be Element of L; :: thesis: for A being non empty set
for B being Finite_Subset of A
for f being Function of A,the carrier of L st B <> {} & ( for x being Element of A st x in B holds
f . x [= u ) holds
FinJoin B,f [= u
let A be non empty set ; :: thesis: for B being Finite_Subset of A
for f being Function of A,the carrier of L st B <> {} & ( for x being Element of A st x in B holds
f . x [= u ) holds
FinJoin B,f [= u
let B be Finite_Subset of A; :: thesis: for f being Function of A,the carrier of L st B <> {} & ( for x being Element of A st x in B holds
f . x [= u ) holds
FinJoin B,f [= u
let f be Function of A,the carrier of L; :: thesis: ( B <> {} & ( for x being Element of A st x in B holds
f . x [= u ) implies FinJoin B,f [= u )
assume that
A1: B <> {} and
A2: for x being Element of A st x in B holds
f . x [= u ; :: thesis: FinJoin B,f [= u
set J = the L_join of L;

A3: now

let x be Element of A; :: thesis: ( x in B implies (the L_join of L [:] f,u) . x = u )
assume x in B ; :: thesis: (the L_join of L [:] f,u) . x = u
then A4: f . x [= u by A2;
thus (the L_join of L [:] f,u) . x = (f . x) "\/" u by FUNCOP_1:60
.= u by A4, LATTICES:def 3 ; :: thesis: verum

end;

the L_join of L is idempotent ;
then (FinJoin B,f) "\/" u = the L_join of L $$ B,(the L_join of L [:] f,u) by A1, Th34, SETWISEO:37
.= u by A1, A3, SETWISEO:33 ;
hence FinJoin B,f [= u by LATTICES:def 3; :: thesis: verum