let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f3 being PartFunc of M,the carrier of V
for f1, f2 being PartFunc of M,COMPLEX holds (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3)
let V be ComplexNormSpace; :: thesis: for f3 being PartFunc of M,the carrier of V
for f1, f2 being PartFunc of M,COMPLEX holds (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3)
let f3 be PartFunc of M,the carrier of V; :: thesis: for f1, f2 being PartFunc of M,COMPLEX holds (f1 - f2) (#) f3 = (f1 (#) f3) - (f2 (#) f3)
let f1, f2 be PartFunc of M,COMPLEX ; :: 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:def 1
.= ((dom f1) /\ (dom f2)) /\ ((dom f3) /\ (dom f3)) by VALUED_1:8
.= (((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 x be Element of M; :: thesis: ( x in dom ((f1 - f2) (#) f3) implies ((f1 - f2) (#) f3) /. x = ((f1 (#) f3) - (f2 (#) f3)) /. x )
assume A2: x in dom ((f1 - f2) (#) f3) ; :: thesis: ((f1 - f2) (#) f3) /. x = ((f1 (#) f3) - (f2 (#) f3)) /. x
then x in (dom (f1 (#) f3)) /\ (dom (f2 (#) f3)) by A1, Def2;
then A3: ( x in dom (f1 (#) f3) & x in dom (f2 (#) f3) ) by XBOOLE_0:def 4;
x in (dom (f1 - f2)) /\ (dom f3) by A2, Def3;
then A4: x in dom (f1 - f2) by XBOOLE_0:def 4;
then A5: x in (dom f1) /\ (dom (- f2)) by VALUED_1:def 1;
then A6: x in dom (- f2) by XBOOLE_0:def 4;
then A7: (- f2) /. x = (- f2) . x by PARTFUN1:def 8;
A8: (- f2) . x = - (f2 . x) by VALUED_1:8;
x in dom f1 by A5, XBOOLE_0:def 4;
then A9: f1 /. x = f1 . x by PARTFUN1:def 8;
dom (- f2) = dom f2 by VALUED_1:8;
then A10: f2 /. x = f2 . x by A6, PARTFUN1:def 8;
(f1 + (- f2)) /. x = (f1 + (- f2)) . x by A4, PARTFUN1:def 8;
then (f1 + (- f2)) /. x = (f1 . x) + ((- f2) . x) by A4, VALUED_1:def 1
.= (f1 /. x) + ((- f2) /. x) by A6, A9, PARTFUN1:def 8 ;
hence ((f1 - f2) (#) f3) /. x = ((f1 /. x) - (f2 /. x)) * (f3 /. x) by A2, A7, A8, A10, Def3
.= ((f1 /. x) * (f3 /. x)) - ((f2 /. x) * (f3 /. x)) by CLVECT_1:11
.= ((f1 (#) f3) /. x) - ((f2 /. x) * (f3 /. x)) by A3, Def3
.= ((f1 (#) f3) /. x) - ((f2 (#) f3) /. x) by A3, Def3
.= ((f1 (#) f3) - (f2 (#) f3)) /. x by A1, A2, Def2 ;
:: thesis: verum

end;

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