let f1, f2, f3 be complex-valued Function; :: thesis: (f1 + f2) (#) f3 = (f1 (#) f3) + (f2 (#) f3)
A1: dom ((f1 + f2) (#) f3) = (dom (f1 + f2)) /\ (dom f3) by VALUED_1:def 4
.= ((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 VALUED_1:def 4
.= (dom (f1 (#) f3)) /\ (dom (f2 (#) f3)) by VALUED_1:def 4
.= dom ((f1 (#) f3) + (f2 (#) f3)) by VALUED_1:def 1 ;

now :: thesis: for c being object st c in dom ((f1 + f2) (#) f3) holds
((f1 + f2) (#) f3) . c = ((f1 (#) f3) + (f2 (#) f3)) . c

let c be object ; :: 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 VALUED_1:def 4;
then A3: c in dom (f1 + f2) by XBOOLE_0:def 4;
thus ((f1 + f2) (#) f3) . c = ((f1 + f2) . c) * (f3 . c) by VALUED_1:5
.= ((f1 . c) + (f2 . c)) * (f3 . c) by A3, VALUED_1:def 1
.= ((f1 . c) * (f3 . c)) + ((f2 . c) * (f3 . c))
.= ((f1 (#) f3) . c) + ((f2 . c) * (f3 . c)) by VALUED_1:5
.= ((f1 (#) f3) . c) + ((f2 (#) f3) . c) by VALUED_1:5
.= ((f1 (#) f3) + (f2 (#) f3)) . c by A1, A2, VALUED_1:def 1 ; :: thesis: verum

end;

hence (f1 + f2) (#) f3 = (f1 (#) f3) + (f2 (#) f3) by A1, FUNCT_1:2; :: thesis: verum