Are you sure it has no restrictions on A?
If not, then if A=0, then B not necessarily = C. For example, 0*2 = 0*4, 4 not equal 2.
tripod87, he said "Cartesian products". A, B, and C are sets, not numbers.
However, with slightly different notation, the same objection works! (So it is possible that tripod87 meant that but used numerical notation.) If A is the empty set then A X B and A X C are empty for all B and C.
Suppose A is not empty. If x is any member of B, there exist a pair (a, x) in A X B for some a in A. Since A X B= A X C, that pair is also in A x C. Therefore x is in C. Then B is a subset of C. Can you finish?