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) = x ) & dom f2 = [:X,Y:] & ( for x, y being object st x in X & y in Y holds
f2 . (x,y) = x ) implies f1 = f2 )
assume that
A1: dom f1 = [:X,Y:] and
A2: for x, y being object st x in X & y in Y holds
f1 . (x,y) = x and
A3: dom f2 = [:X,Y:] and
A4: for x, y being object st x in X & y in Y holds
f2 . (x,y) = x ; :: 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 A5: ( x in X & y in Y ) ; :: thesis: f1 . (x,y) = f2 . (x,y)
then f1 . (x,y) = x by A2;
hence f1 . (x,y) = f2 . (x,y) by A4, A5; :: thesis: verum

end;

hence f1 = f2 by A1, A3, Th6; :: thesis: verum