let f1, f2, f3 be complex-valued Function; :: thesis: f1 + (f2 - f3) = (f1 + f2) - f3
A1: dom (f1 + (f2 - f3)) = (dom f1) /\ (dom (f2 - f3)) by VALUED_1:def 1
.= (dom f1) /\ ((dom f2) /\ (dom f3)) by VALUED_1:12
.= ((dom f1) /\ (dom f2)) /\ (dom f3) by XBOOLE_1:16
.= (dom (f1 + f2)) /\ (dom f3) by VALUED_1:def 1
.= dom ((f1 + f2) - f3) by VALUED_1:12 ;

now :: thesis: for c being object st c in dom (f1 + (f2 - f3)) holds
(f1 + (f2 - f3)) . c = ((f1 + f2) - f3) . c

let c be object ; :: thesis: ( c in dom (f1 + (f2 - f3)) implies (f1 + (f2 - f3)) . c = ((f1 + f2) - f3) . c )
assume A2: c in dom (f1 + (f2 - f3)) ; :: thesis: (f1 + (f2 - f3)) . c = ((f1 + f2) - f3) . c
then c in (dom f1) /\ (dom (f2 - f3)) by VALUED_1:def 1;
then A3: c in dom (f2 - f3) by XBOOLE_0:def 4;
c in (dom (f1 + f2)) /\ (dom f3) by A1, A2, VALUED_1:12;
then A4: c in dom (f1 + f2) by XBOOLE_0:def 4;
thus (f1 + (f2 - f3)) . c = (f1 . c) + ((f2 - f3) . c) by A2, VALUED_1:def 1
.= (f1 . c) + ((f2 . c) - (f3 . c)) by A3, VALUED_1:13
.= ((f1 . c) + (f2 . c)) - (f3 . c)
.= ((f1 + f2) . c) - (f3 . c) by A4, VALUED_1:def 1
.= ((f1 + f2) - f3) . c by A1, A2, VALUED_1:13 ; :: thesis: verum

end;

hence f1 + (f2 - f3) = (f1 + f2) - f3 by A1, FUNCT_1:2; :: thesis: verum