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

now

let x be set ; :: thesis: ( x in (dom f) /\ (dom g) implies f1 . x = f2 . x )
assume x in (dom f) /\ (dom g) ; :: thesis: f1 . x = f2 . x
then ( f1 . x = (f . x) /\ (g . x) & f2 . x = (f . x) /\ (g . x) ) by A2, A4;
hence f1 . x = f2 . x ; :: thesis: verum

end;

hence f1 = f2 by A1, A3, FUNCT_1:9; :: thesis: verum