let C be non empty set ; :: thesis: for r being Real
for V being non empty scalar-associative RLSStruct
for f1 being PartFunc of C,REAL
for f2 being PartFunc of C,V holds r (#) (f1 (#) f2) = (r (#) f1) (#) f2
let r be Real; :: thesis: for V being non empty scalar-associative RLSStruct
for f1 being PartFunc of C,REAL
for f2 being PartFunc of C,V holds r (#) (f1 (#) f2) = (r (#) f1) (#) f2
let V be non empty scalar-associative RLSStruct ; :: thesis: for f1 being PartFunc of C,REAL
for f2 being PartFunc of C,V holds r (#) (f1 (#) f2) = (r (#) f1) (#) f2
let f1 be PartFunc of C,REAL; :: thesis: for f2 being PartFunc of C,V holds r (#) (f1 (#) f2) = (r (#) f1) (#) f2
let f2 be PartFunc of C,V; :: thesis: r (#) (f1 (#) f2) = (r (#) f1) (#) f2
A1: dom (r (#) (f1 (#) f2)) = dom (f1 (#) f2) by Def4
.= (dom f1) /\ (dom f2) by Def3
.= (dom (r (#) f1)) /\ (dom f2) by VALUED_1:def 5
.= dom ((r (#) f1) (#) f2) by Def3 ;

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

let c be Element of C; :: thesis: ( c in dom (r (#) (f1 (#) f2)) implies (r (#) (f1 (#) f2)) /. c = ((r (#) f1) (#) f2) /. c )
assume A2: c in dom (r (#) (f1 (#) f2)) ; :: thesis: (r (#) (f1 (#) f2)) /. c = ((r (#) f1) (#) f2) /. c
then A3: c in dom (f1 (#) f2) by Def4;
c in (dom (r (#) f1)) /\ (dom f2) by A1, A2, Def3;
then A4: c in dom (r (#) f1) 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
.= (r * (f1 . c)) * (f2 /. c) by RLVECT_1:def 7
.= ((r (#) f1) . c) * (f2 /. c) by A4, VALUED_1:def 5
.= ((r (#) f1) (#) f2) /. c by A1, A2, Def3 ; :: thesis: verum

end;

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