let f1, f2 be Function; :: thesis: ( dom f1 = dom f & ( for x being object st x in dom f holds
f1 . x = Funcs ((f . x),X) ) & dom f2 = dom f & ( for x being object st x in dom f holds
f2 . x = Funcs ((f . x),X) ) implies f1 = f2 )
assume that
A2: dom f1 = dom f and
A3: for x being object st x in dom f holds
f1 . x = Funcs ((f . x),X) and
A4: dom f2 = dom f and
A5: for x being object st x in dom f holds
f2 . x = Funcs ((f . x),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 A6: x in dom f ; :: thesis: f1 . x = f2 . x
then f1 . x = Funcs ((f . x),X) by A3;
hence f1 . x = f2 . x by A5, A6; :: thesis: verum

end;

hence f1 = f2 by A2, A4; :: thesis: verum