let X1, X2 be Subset of Q; :: thesis: ( ( for z being Element of Q holds
( z in X1 iff for x, y being Element of Q holds (x * y) * z = x * (y * z) ) ) & ( for z being Element of Q holds
( z in X2 iff for x, y being Element of Q holds (x * y) * z = x * (y * z) ) ) implies X1 = X2 )
assume that
A6: for x being Element of Q holds
( x in X1 iff S₃[x] ) and
A7: for x being Element of Q holds
( x in X2 iff S₃[x] ) ; :: thesis: X1 = X2
thus X1 = X2 from SUBSET_1:sch 2(A6, A7); :: thesis: verum