let A1, A2 be strict LattStr ; :: thesis: ( the carrier of A1 = RoughSets X & ( for A, B being Element of RoughSets X
for A9, B9 being RoughSet of X st A = A9 & B = B9 holds
( the L_join of A1 . A,B = A9 _\/_ B9 & the L_meet of A1 . A,B = A9 _/\_ B9 ) ) & the carrier of A2 = RoughSets X & ( for A, B being Element of RoughSets X
for A9, B9 being RoughSet of X st A = A9 & B = B9 holds
( the L_join of A2 . A,B = A9 _\/_ B9 & the L_meet of A2 . A,B = A9 _/\_ B9 ) ) implies A1 = A2 )
assume that
A3: ( the carrier of A1 = RoughSets X & ( for A, B being Element of RoughSets X
for A9, B9 being RoughSet of X st A = A9 & B = B9 holds
( the L_join of A1 . A,B = A9 _\/_ B9 & the L_meet of A1 . A,B = A9 _/\_ B9 ) ) ) and
A4: ( the carrier of A2 = RoughSets X & ( for A, B being Element of RoughSets X
for A9, B9 being RoughSet of X st A = A9 & B = B9 holds
( the L_join of A2 . A,B = A9 _\/_ B9 & the L_meet of A2 . A,B = A9 _/\_ B9 ) ) ) ; :: thesis: A1 = A2
reconsider a3 = the L_meet of A1, a4 = the L_meet of A2, a1 = the L_join of A1, a2 = the L_join of A2 as BinOp of (RoughSets X) by A3, A4;

now

let x, y be Element of RoughSets X; :: thesis: a1 . x,y = a2 . x,y
thus a1 . x,y = (@ x) _\/_ (@ y) by A3
.= a2 . x,y by A4 ; :: thesis: verum

end;

then A5: a1 = a2 by BINOP_1:2;

now

let x, y be Element of RoughSets X; :: thesis: a3 . x,y = a4 . x,y
thus a3 . x,y = (@ x) _/\_ (@ y) by A3
.= a4 . x,y by A4 ; :: thesis: verum

end;

hence A1 = A2 by A3, A4, A5, BINOP_1:2; :: thesis: verum