Prove if the sets A x B = A x C and A is nonempty then B=C

Prove if the sets A x B = A x C and A is nonempty then B=C

Suppose there exists an element b in the set B and an element a in the set A. Then a and b are elements in A x B. But since A x B = A x C then a and b are also elements in A x C. So b is an element in C, which proves B is a subset of C. You could prove that C is a subset of B in a similar manner and so B = C.