let C be non empty set ; :: thesis: for f1, f2, f3 being PartFunc of C,COMPLEX holds (f1 + f2) (#) f3 = (f1 (#) f3) + (f2 (#) f3)
let f1, f2, f3 be PartFunc of C,COMPLEX; :: thesis: (f1 + f2) (#) f3 = (f1 (#) f3) + (f2 (#) f3)
A1: dom ((f1 + f2) (#) f3) = (dom (f1 + f2)) /\ (dom f3) by Th5
.= ((dom f1) /\ (dom f2)) /\ ((dom f3) /\ (dom f3)) by VALUED_1:def 1
.= (((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 Th5
.= (dom (f1 (#) f3)) /\ (dom (f2 (#) f3)) by Th5
.= dom ((f1 (#) f3) + (f2 (#) f3)) by VALUED_1:def 1 ;

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 A3: c in (dom (f1 (#) f3)) /\ (dom (f2 (#) f3)) by A1, VALUED_1:def 1;
then A4: c in dom (f1 (#) f3) by XBOOLE_0:def 4;
c in (dom (f1 + f2)) /\ (dom f3) by A2, Th5;
then A5: c in dom (f1 + f2) by XBOOLE_0:def 4;
A6: c in dom (f2 (#) f3) by A3, XBOOLE_0:def 4;
thus ((f1 + f2) (#) f3) /. c = ((f1 + f2) /. c) * (f3 /. c) by A2, Th5
.= ((f1 /. c) + (f2 /. c)) * (f3 /. c) by A5, Th3
.= ((f1 /. c) * (f3 /. c)) + ((f2 /. c) * (f3 /. c))
.= ((f1 (#) f3) /. c) + ((f2 /. c) * (f3 /. c)) by A4, Th5
.= ((f1 (#) f3) /. c) + ((f2 (#) f3) /. c) by A6, Th5
.= ((f1 (#) f3) + (f2 (#) f3)) /. c by A1, A2, Th3 ; :: thesis: verum

end;

hence (f1 + f2) (#) f3 = (f1 (#) f3) + (f2 (#) f3) by A1, PARTFUN2:1; :: thesis: verum