let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f2 being PartFunc of M,the carrier of V
for X, Y being set
for f1 being PartFunc of M,COMPLEX st f1 | X is constant & f2 | Y is constant holds
(f1 (#) f2) | (X /\ Y) is constant
let V be ComplexNormSpace; :: thesis: for f2 being PartFunc of M,the carrier of V
for X, Y being set
for f1 being PartFunc of M,COMPLEX st f1 | X is constant & f2 | Y is constant holds
(f1 (#) f2) | (X /\ Y) is constant
let f2 be PartFunc of M,the carrier of V; :: thesis: for X, Y being set
for f1 being PartFunc of M,COMPLEX st f1 | X is constant & f2 | Y is constant holds
(f1 (#) f2) | (X /\ Y) is constant
let X, Y be set ; :: thesis: for f1 being PartFunc of M,COMPLEX st f1 | X is constant & f2 | Y is constant holds
(f1 (#) f2) | (X /\ Y) is constant
let f1 be PartFunc of M,COMPLEX ; :: thesis: ( f1 | X is constant & f2 | Y is constant implies (f1 (#) f2) | (X /\ Y) is constant )
assume A1: ( f1 | X is constant & f2 | Y is constant ) ; :: thesis: (f1 (#) f2) | (X /\ Y) is constant
then consider z1 being Complex such that
A2: for c being Element of M st c in X /\ (dom f1) holds
f1 . c = z1 by PARTFUN2:76;
consider r2 being VECTOR of V such that
A3: for c being Element of M st c in Y /\ (dom f2) holds
f2 /. c = r2 by A1, PARTFUN2:54;

now

let c be Element of M; :: thesis: ( c in (X /\ Y) /\ (dom (f1 (#) f2)) implies (f1 (#) f2) /. c = z1 * r2 )
assume c in (X /\ Y) /\ (dom (f1 (#) f2)) ; :: thesis: (f1 (#) f2) /. c = z1 * r2
then A4: ( c in X /\ Y & c in dom (f1 (#) f2) ) by XBOOLE_0:def 4;
then A5: ( c in X & c in Y & c in (dom f1) /\ (dom f2) ) by Def3, XBOOLE_0:def 4;
then A6: ( c in dom f1 & c in dom f2 ) by XBOOLE_0:def 4;
then A7: ( c in X /\ (dom f1) & c in Y /\ (dom f2) ) by A5, XBOOLE_0:def 4;
A8: f1 /. c = f1 . c by A6, PARTFUN1:def 8;
thus (f1 (#) f2) /. c = (f1 /. c) * (f2 /. c) by A4, Def3
.= z1 * (f2 /. c) by A2, A7, A8
.= z1 * r2 by A3, A7 ; :: thesis: verum

end;

hence (f1 (#) f2) | (X /\ Y) is constant by PARTFUN2:54; :: thesis: verum