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

proof

let x be object ; :: thesis: ( x in (dom f) /\ (dom g) implies f1 . x = f2 . x )
assume A5: x in (dom f) /\ (dom g) ; :: thesis: f1 . x = f2 . x
then f1 . x = [(f . x),(g . x)] by A1, A2;
hence f1 . x = f2 . x by A3, A4, A5; :: thesis: verum

end;

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