let X1, X2 be Subset of Q; :: thesis: ( ( for x being Element of Q holds
( x in X1 iff for y, z being Element of Q holds (x * y) * z = x * (y * z) ) ) & ( for x being Element of Q holds
( x in X2 iff for y, z being Element of Q holds (x * y) * z = x * (y * z) ) ) implies X1 = X2 )
assume that
A1: for x being Element of Q holds
( x in X1 iff S₁[x] ) and
A2: for x being Element of Q holds
( x in X2 iff S₁[x] ) ; :: thesis: X1 = X2
thus X1 = X2 from SUBSET_1:sch 2(A1, A2); :: thesis: verum