I need help proving something in set theory...?

Question

Prove: If A, B, C are sets, then (A-B) X C = (AXC) - (BXC), where "A-B" is the set containing all elements of A that are not in B, and AXC denotes the Cartesian product between A and C. Obviously, to prove this, one must show that for any ordered pair (x,y), if it is contained in the set on the left side of the equation, it is necessarily in the one on the right, and vice versa. The proof seems almost obvious. I just want someone to write down a set of logical steps showing how they arrive at the above statement.

Answer 1

let (x,y) element of (A-B)XC
so, x element of (A-B) AND y elemnt of C
so x element of A AND x not element of B AND y element of C
so (x element of A AND y element of C) AND (x not element of B)
so (x,y) element of (AXC) AND (x,y) not elelmet of (BXC) ****
so (x,y) element of (AXC) - (BXC)

First part is done.

noe, let,
(x,y) element of (AXC)-(BXC)
so (x,y) element of (AXC) AND NOT ( (x,y) element of( BXC))
so x element of A AND y element of C AND NOT (x element of B AND y element of C)
so x element of A AND y element of C AND ( x not element of B OR y not element of C)
so x element of A AND y element of C AND x not element of B ####
so (x,y) element of (A-B)XC

Now if you have confusion anywhere that is probably the lines marked with * and #.
Lets dicuss this
*******
if x is not element of be it can not be element of BXC, whtever happens to y and C.

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This should be our ststement OR some other statement.
But that statement is contradiction, so we may ignore that part.