Let K1 over F and K2 over F be two finite field extensions with K1 and K2 subfield of L..
L over F is Algebraic extension and K1K2 is the smallest subfield of L generated by k1 and K2..
Suppose K1 is splitting of f1(x) in F[x] over F and K2 is splitting of f2(x) in F[X] over F..

Then why is K1K2 splitting field of f1(x)f2(x) in F[X] over F???