let f1, f2 be Function; :: thesis: ( dom f1 = [:X,Y:] & ( for x, y being object st x in X & y in Y holds
f1 . (x,y) = y ) & dom f2 = [:X,Y:] & ( for x, y being object st x in X & y in Y holds
f2 . (x,y) = y ) implies f1 = f2 )
assume that
A6: dom f1 = [:X,Y:] and
A7: for x, y being object st x in X & y in Y holds
f1 . (x,y) = y and
A8: dom f2 = [:X,Y:] and
A9: for x, y being object st x in X & y in Y holds
f2 . (x,y) = y ; :: thesis: f1 = f2
for x, y being object st x in X & y in Y holds
f1 . (x,y) = f2 . (x,y)

proof

let x, y be object ; :: thesis: ( x in X & y in Y implies f1 . (x,y) = f2 . (x,y) )
assume A10: ( x in X & y in Y ) ; :: thesis: f1 . (x,y) = f2 . (x,y)
then f1 . (x,y) = y by A7;
hence f1 . (x,y) = f2 . (x,y) by A9, A10; :: thesis: verum

end;

hence f1 = f2 by A6, A8, Th6; :: thesis: verum