Could someone please check my work?

Prove ( AnB ) x C = (A X C ) n ( B XC ) Fill in The blanks. Let ( x, y ) E ( AnB) X C. Then XE AnB and yE C. Since X E AnB, XE A and XEB. Thus, ( x,y) E A XC and (x, y ) E B X C. Hence ( x,y ) E ( A XC ) n (BX C), So ( AnB) x C S ( AXC)n( BXC ). On The other hand, suppose That ( x, y) E ( AXC) n (BXC ). Then ( x , y ) E A X C and (x,y ) E B X C. Since (x,y) EAXC , XE A and y EC Since (x, y ) EB X C , XEB and yEc. Thus, X E AnB, So ( x, y ) E ( AnB) XC and (AXC) A ( B X C) S ( AnB) XC