Could someone please tell me if my proof is correct

Suppose f: A B. Let C, and Cz be subsars of A. Prove That f (c, ) of(ca) s f (c, UC2). Ler y E f ( c, ) Uf (c2). = > yef (c ) or yef(c 2 ) - F X E C such That f ( x)= y or ] x EC, such Ther f(x)=y = > since XEC , or X E C 2 X E C , UC , 7 4 = f ( x ) E f (C, UC2) -> Since yef ( c, ) of ( ca) and yef ( G UC). F CC, ) U f ( C, ) = f (c, UC2)