Prove that A X (B U C) = (A X B) U (A X C)

Prove that A X (B U C) = (A X B) U (A X C)

In English this is saying to prove that the Cartesian product of A and B union C is equal to the Cartesian product of A and B union the Cartesian product of A union C.

iff=if and only iff

Consider that the ordered pair (x,y) is an element of A X (B U C)
which is true iff x is an element of A and y is an element on (B U C)
iff x is an element of A and y is an element of B or y is an element of C
iff x is in A and y is in B or x is in A and y is in C
iff (x,y) is in A X B or (x,y) is in A X C
iff (x,y) is in (A X B) U (A X C)

thus A X (B U C) = (A X B) U (A X C)