let C be non empty set ; :: thesis: for V being RealNormSpace
for f2 being PartFunc of C,the carrier of V
for r being Real
for f1 being PartFunc of C,REAL holds r (#) (f1 (#) f2) = f1 (#) (r (#) f2)
let V be RealNormSpace; :: thesis: for f2 being PartFunc of C,the carrier of V
for r being Real
for f1 being PartFunc of C,REAL holds r (#) (f1 (#) f2) = f1 (#) (r (#) f2)
let f2 be PartFunc of C,the carrier of V; :: thesis: for r being Real
for f1 being PartFunc of C,REAL holds r (#) (f1 (#) f2) = f1 (#) (r (#) f2)
let r be Real; :: thesis: for f1 being PartFunc of C,REAL holds r (#) (f1 (#) f2) = f1 (#) (r (#) f2)
let f1 be PartFunc of C,REAL ; :: thesis: r (#) (f1 (#) f2) = f1 (#) (r (#) f2)
A1: dom (r (#) (f1 (#) f2)) = dom (f1 (#) f2) by Def4
.= (dom f1) /\ (dom f2) by Def3
.= (dom f1) /\ (dom (r (#) f2)) by Def4
.= dom (f1 (#) (r (#) f2)) by Def3 ;

now

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 Def4;
c in (dom f1) /\ (dom (r (#) f2)) by A1, A2, Def3;
then A4: c in dom (r (#) f2) by XBOOLE_0:def 4;
thus (r (#) (f1 (#) f2)) /. c = r * ((f1 (#) f2) /. c) by A2, Def4
.= r * ((f1 . c) * (f2 /. c)) by A3, Def3
.= ((f1 . c) * r) * (f2 /. c) by RLVECT_1:def 9, RLVECT_1:def 10, RLVECT_1:def 11
.= (f1 . c) * (r * (f2 /. c)) by RLVECT_1:def 9, RLVECT_1:def 10, RLVECT_1:def 11
.= (f1 . c) * ((r (#) f2) /. c) by A4, Def4
.= (f1 (#) (r (#) f2)) /. c by A1, A2, Def3 ; :: thesis: verum

end;

hence r (#) (f1 (#) f2) = f1 (#) (r (#) f2) by A1, PARTFUN2:3; :: thesis: verum