let C be non empty set ; :: thesis: for V being RealNormSpace
for f1, f2 being PartFunc of C,the carrier of V
for r being Real holds r (#) (f1 + f2) = (r (#) f1) + (r (#) f2)
let V be RealNormSpace; :: thesis: for f1, f2 being PartFunc of C,the carrier of V
for r being Real holds r (#) (f1 + f2) = (r (#) f1) + (r (#) f2)
let f1, f2 be PartFunc of C,the carrier of V; :: thesis: for r being Real holds r (#) (f1 + f2) = (r (#) f1) + (r (#) f2)
let r be Real; :: thesis: r (#) (f1 + f2) = (r (#) f1) + (r (#) f2)
A1: dom (r (#) (f1 + f2)) = dom (f1 + f2) by Def4
.= (dom f1) /\ (dom f2) by Def1
.= (dom f1) /\ (dom (r (#) f2)) by Def4
.= (dom (r (#) f1)) /\ (dom (r (#) f2)) by Def4
.= dom ((r (#) f1) + (r (#) f2)) by Def1 ;

now

let c be Element of C; :: 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 Def4;
A4: c in (dom (r (#) f1)) /\ (dom (r (#) f2)) by A1, A2, Def1;
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, Def4
.= r * ((f1 /. c) + (f2 /. c)) by A3, Def1
.= (r * (f1 /. c)) + (r * (f2 /. c)) by RLVECT_1:def 9, RLVECT_1:def 10, RLVECT_1:def 11, RLVECT_1:def 8
.= ((r (#) f1) /. c) + (r * (f2 /. c)) by A5, Def4
.= ((r (#) f1) /. c) + ((r (#) f2) /. c) by A6, Def4
.= ((r (#) f1) + (r (#) f2)) /. c by A1, A2, Def1 ; :: thesis: verum

end;

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