let f1, f2 be complex-valued Function; :: thesis: for r being Complex holds r (#) (f1 - f2) = (r (#) f1) - (r (#) f2)
let r be Complex; :: thesis: r (#) (f1 - f2) = (r (#) f1) - (r (#) f2)
A1: dom (r (#) (f1 - f2)) = dom (f1 - f2) by VALUED_1:def 5
.= (dom f1) /\ (dom f2) by VALUED_1:12
.= (dom f1) /\ (dom (r (#) f2)) by VALUED_1:def 5
.= (dom (r (#) f1)) /\ (dom (r (#) f2)) by VALUED_1:def 5
.= dom ((r (#) f1) - (r (#) f2)) by VALUED_1:12 ;

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

let c be object ; :: thesis: ( c in dom (r (#) (f1 - f2)) implies (r (#) (f1 - f2)) . c = ((r (#) f1) - (r (#) f2)) . c )
assume A2: c in dom (r (#) (f1 - f2)) ; :: thesis: (r (#) (f1 - f2)) . c = ((r (#) f1) - (r (#) f2)) . c
then A3: c in dom (f1 - f2) by VALUED_1:def 5;
A4: c in (dom (r (#) f1)) /\ (dom (r (#) f2)) by A1, A2, VALUED_1:12;
then A5: c in dom (r (#) f1) by XBOOLE_0:def 4;
A6: c in dom (r (#) f2) by A4, XBOOLE_0:def 4;
thus (r (#) (f1 - f2)) . c = r * ((f1 - f2) . c) by A2, VALUED_1:def 5
.= r * ((f1 . c) - (f2 . c)) by A3, VALUED_1:13
.= (r * (f1 . c)) - (r * (f2 . c))
.= ((r (#) f1) . c) - (r * (f2 . c)) by A5, VALUED_1:def 5
.= ((r (#) f1) . c) - ((r (#) f2) . c) by A6, VALUED_1:def 5
.= ((r (#) f1) - (r (#) f2)) . c by A1, A2, VALUED_1:13 ; :: thesis: verum

end;

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