let C be non empty set ; :: thesis: for f1, f2 being PartFunc of C,COMPLEX
for r being Complex holds r (#) (f1 (#) f2) = f1 (#) (r (#) f2)
let f1, f2 be PartFunc of C,COMPLEX; :: thesis: for r being Complex holds r (#) (f1 (#) f2) = f1 (#) (r (#) f2)
let r be Complex; :: thesis: r (#) (f1 (#) f2) = f1 (#) (r (#) f2)
A1: dom (r (#) (f1 (#) f2)) = dom (f1 (#) f2) by Th4
.= (dom f1) /\ (dom f2) by Th3
.= (dom f1) /\ (dom (r (#) f2)) by Th4
.= dom (f1 (#) (r (#) f2)) by Th3 ;

now :: thesis: for c being Element of C st c in dom (r (#) (f1 (#) f2)) holds
(r (#) (f1 (#) f2)) /. c = (f1 (#) (r (#) f2)) /. c

let c be Element of C; :: thesis: ( c in dom (r (#) (f1 (#) f2)) implies (r (#) (f1 (#) f2)) /. c = (f1 (#) (r (#) f2)) /. c )
assume A2: c in dom (r (#) (f1 (#) f2)) ; :: thesis: (r (#) (f1 (#) f2)) /. c = (f1 (#) (r (#) f2)) /. c
then A3: c in dom (f1 (#) f2) by Th4;
c in (dom f1) /\ (dom (r (#) f2)) by A1, A2, Th3;
then A4: c in dom (r (#) f2) by XBOOLE_0:def 4;
thus (r (#) (f1 (#) f2)) /. c = r * ((f1 (#) f2) /. c) by A2, Th4
.= r * ((f1 /. c) * (f2 /. c)) by A3, Th3
.= (f1 /. c) * (r * (f2 /. c))
.= (f1 /. c) * ((r (#) f2) /. c) by A4, Th4
.= (f1 (#) (r (#) f2)) /. c by A1, A2, Th3 ; :: thesis: verum

end;

hence r (#) (f1 (#) f2) = f1 (#) (r (#) f2) by A1, PARTFUN2:1; :: thesis: verum