let X, Y be set ; :: thesis: for C being non empty set
for V being RealNormSpace
for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL st f1 | X is V8() & f2 | Y is V8() holds
(f1 (#) f2) | (X /\ Y) is V8()
let C be non empty set ; :: thesis: for V being RealNormSpace
for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL st f1 | X is V8() & f2 | Y is V8() holds
(f1 (#) f2) | (X /\ Y) is V8()
let V be RealNormSpace; :: thesis: for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL st f1 | X is V8() & f2 | Y is V8() holds
(f1 (#) f2) | (X /\ Y) is V8()
let f2 be PartFunc of C,V; :: thesis: for f1 being PartFunc of C,REAL st f1 | X is V8() & f2 | Y is V8() holds
(f1 (#) f2) | (X /\ Y) is V8()
let f1 be PartFunc of C,REAL; :: thesis: ( f1 | X is V8() & f2 | Y is V8() implies (f1 (#) f2) | (X /\ Y) is V8() )
assume that
A1: f1 | X is V8() and
A2: f2 | Y is V8() ; :: thesis: (f1 (#) f2) | (X /\ Y) is V8()
consider r1 being Real such that
A3: for c being Element of C st c in X /\ (dom f1) holds
f1 . c = r1 by A1, PARTFUN2:57;
consider r2 being VECTOR of V such that
A4: for c being Element of C st c in Y /\ (dom f2) holds
f2 /. c = r2 by A2, PARTFUN2:35;

now

let c be Element of C; :: thesis: ( c in (X /\ Y) /\ (dom (f1 (#) f2)) implies (f1 (#) f2) /. c = r1 * r2 )
assume A5: c in (X /\ Y) /\ (dom (f1 (#) f2)) ; :: thesis: (f1 (#) f2) /. c = r1 * r2
then A6: c in X /\ Y by XBOOLE_0:def 4;
then A7: c in X by XBOOLE_0:def 4;
A8: c in dom (f1 (#) f2) by A5, XBOOLE_0:def 4;
then A9: c in (dom f1) /\ (dom f2) by Def3;
then c in dom f1 by XBOOLE_0:def 4;
then A10: c in X /\ (dom f1) by A7, XBOOLE_0:def 4;
A11: c in Y by A6, XBOOLE_0:def 4;
c in dom f2 by A9, XBOOLE_0:def 4;
then A12: c in Y /\ (dom f2) by A11, XBOOLE_0:def 4;
thus (f1 (#) f2) /. c = (f1 . c) * (f2 /. c) by A8, Def3
.= r1 * (f2 /. c) by A3, A10
.= r1 * r2 by A4, A12 ; :: thesis: verum

end;

hence (f1 (#) f2) | (X /\ Y) is V8() by PARTFUN2:35; :: thesis: verum