let f1, f2, f3 be complex-valued Function; :: thesis: f1 - (f2 + f3) = (f1 - f2) - f3
thus A1: dom (f1 - (f2 + f3)) = (dom f1) /\ (dom (f2 + f3)) by VALUED_1:12
.= (dom f1) /\ ((dom f2) /\ (dom f3)) by VALUED_1:def 1
.= ((dom f1) /\ (dom f2)) /\ (dom f3) by XBOOLE_1:16
.= (dom (f1 - f2)) /\ (dom f3) by VALUED_1:12
.= dom ((f1 - f2) - f3) by VALUED_1:12 ; :: according to FUNCT_1:def 11 :: thesis: for b₁ being object holds
( not b₁ in dom (f1 - (f2 + f3)) or (f1 - (f2 + f3)) . b₁ = ((f1 - f2) - f3) . b₁ )
let c be object ; :: thesis: ( not c in dom (f1 - (f2 + f3)) or (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_1:12;
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_1:13
.= (f1 . c) - ((f2 . c) + (f3 . c)) by A3, VALUED_1:def 1
.= ((f1 . c) - (f2 . c)) - (f3 . c)
.= ((f1 - f2) . c) - (f3 . c) by A4, VALUED_1:13
.= ((f1 - f2) - f3) . c by A1, A2, VALUED_1:13 ; :: thesis: verum