let C be non empty set ; :: thesis: for V being RealNormSpace
for f3 being PartFunc of C,the carrier of V
for f1, f2 being PartFunc of C,REAL holds (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3)
let V be RealNormSpace; :: thesis: for f3 being PartFunc of C,the carrier of V
for f1, f2 being PartFunc of C,REAL holds (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3)
let f3 be PartFunc of C,the carrier of V; :: thesis: for f1, f2 being PartFunc of C,REAL holds (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3)
let f1, f2 be PartFunc of C,REAL ; :: thesis: (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3)
A1: dom ((f1 - f2) (#) f3) = (dom (f1 - f2)) /\ (dom f3) by Def3
.= ((dom f1) /\ (dom f2)) /\ ((dom f3) /\ (dom f3)) by VALUED_1:12
.= (((dom f1) /\ (dom f2)) /\ (dom f3)) /\ (dom f3) by XBOOLE_1:16
.= (((dom f1) /\ (dom f3)) /\ (dom f2)) /\ (dom f3) by XBOOLE_1:16
.= ((dom f1) /\ (dom f3)) /\ ((dom f2) /\ (dom f3)) by XBOOLE_1:16
.= (dom (f1 (#) f3)) /\ ((dom f2) /\ (dom f3)) by Def3
.= (dom (f1 (#) f3)) /\ (dom (f2 (#) f3)) by Def3
.= dom ((f1 (#) f3) - (f2 (#) f3)) by Def2 ;

now

let c be Element of C; :: thesis: ( c in dom ((f1 - f2) (#) f3) implies ((f1 - f2) (#) f3) /. c = ((f1 (#) f3) - (f2 (#) f3)) /. c )
assume A2: c in dom ((f1 - f2) (#) f3) ; :: thesis: ((f1 - f2) (#) f3) /. c = ((f1 (#) f3) - (f2 (#) f3)) /. c
then c in (dom (f1 - f2)) /\ (dom f3) by Def3;
then A3: c in dom (f1 - f2) by XBOOLE_0:def 4;
A4: c in (dom (f1 (#) f3)) /\ (dom (f2 (#) f3)) by A1, A2, Def2;
then A5: c in dom (f1 (#) f3) by XBOOLE_0:def 4;
A6: c in dom (f2 (#) f3) by A4, XBOOLE_0:def 4;
thus ((f1 - f2) (#) f3) /. c = ((f1 - f2) . c) * (f3 /. c) by A2, Def3
.= ((f1 . c) - (f2 . c)) * (f3 /. c) by A3, VALUED_1:13
.= ((f1 . c) * (f3 /. c)) - ((f2 . c) * (f3 /. c)) by RLVECT_1:49
.= ((f1 (#) f3) /. c) - ((f2 . c) * (f3 /. c)) by A5, Def3
.= ((f1 (#) f3) /. c) - ((f2 (#) f3) /. c) by A6, Def3
.= ((f1 (#) f3) - (f2 (#) f3)) /. c by A1, A2, Def2 ; :: thesis: verum

end;

hence (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3) by A1, PARTFUN2:3; :: thesis: verum