let f1, f2 be Function; :: thesis: ( dom f1 = [:X,Y:] & ( for x, y being set st x in X & y in Y holds
f1 . x,y = x ) & dom f2 = [:X,Y:] & ( for x, y being set 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 set st x in X & y in Y holds
f1 . x,y = x and
A3: dom f2 = [:X,Y:] and
A4: for x, y being set st x in X & y in Y holds
f2 . x,y = x ; :: thesis: f1 = f2
for x, y being set st x in X & y in Y holds
f1 . x,y = f2 . x,y

proof

let x, y be set ; :: thesis: ( x in X & y in Y implies f1 . x,y = f2 . x,y )
assume ( x in X & y in Y ) ; :: thesis: f1 . x,y = f2 . x,y
then ( f1 . x,y = x & f2 . x,y = x ) by A2, A4;
hence f1 . x,y = f2 . x,y ; :: thesis: verum

end;

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