let K, L be non empty LattStr ; :: thesis: ( LattStr(# the carrier of K, the L_join of K, the L_meet of K #) = LattStr(# the carrier of L, the L_join of L, the L_meet of L #) & K is join-commutative implies L is join-commutative )
assume that
A1: LattStr(# the carrier of K, the L_join of K, the L_meet of K #) = LattStr(# the carrier of L, the L_join of L, the L_meet of L #) and
A2: K is join-commutative ; :: thesis: L is join-commutative
L is join-commutative

proof

let x, y be Element of L; :: according to LATTICES:def 4 :: thesis: x |_| y = y |_| x
reconsider x9 = x, y9 = y as Element of K by A1;
x9 |_| y9 = y9 |_| x9 by A2;
hence x |_| y = y |_| x by A1; :: thesis: verum

end;

hence L is join-commutative ; :: thesis: verum