let f1, f2 be Function; :: thesis: ( dom f1 = dom f & ( for x being object st x in dom f holds
f1 . x = Funcs (X,(f . x)) ) & dom f2 = dom f & ( for x being object st x in dom f holds
f2 . x = Funcs (X,(f . x)) ) implies f1 = f2 )
assume that
A1: dom f1 = dom f and
A2: for x being object st x in dom f holds
f1 . x = Funcs (X,(f . x)) and
A3: dom f2 = dom f and
A4: for x being object st x in dom f holds
f2 . x = Funcs (X,(f . x)) ; :: thesis: f1 = f2

now :: thesis: for x being object st x in dom f holds
f1 . x = f2 . x

let x be object ; :: thesis: ( x in dom f implies f1 . x = f2 . x )
assume A5: x in dom f ; :: thesis: f1 . x = f2 . x
then f1 . x = Funcs (X,(f . x)) by A2;
hence f1 . x = f2 . x by A4, A5; :: thesis: verum

end;

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