let X, Y be set ; :: thesis: for C being non empty set
for f1, f2 being PartFunc of C,REAL st f1 | X is constant & f2 | Y is constant holds
( (f1 + f2) | (X /\ Y) is constant & (f1 - f2) | (X /\ Y) is constant & (f1 (#) f2) | (X /\ Y) is constant )
let C be non empty set ; :: thesis: for f1, f2 being PartFunc of C,REAL st f1 | X is constant & f2 | Y is constant holds
( (f1 + f2) | (X /\ Y) is constant & (f1 - f2) | (X /\ Y) is constant & (f1 (#) f2) | (X /\ Y) is constant )
let f1, f2 be PartFunc of C,REAL ; :: thesis: ( f1 | X is constant & f2 | Y is constant implies ( (f1 + f2) | (X /\ Y) is constant & (f1 - f2) | (X /\ Y) is constant & (f1 (#) f2) | (X /\ Y) is constant ) )
assume A1: ( f1 | X is constant & f2 | Y is constant ) ; :: thesis: ( (f1 + f2) | (X /\ Y) is constant & (f1 - f2) | (X /\ Y) is constant & (f1 (#) f2) | (X /\ Y) is constant )
then consider r1 being Real such that
A2: for c being Element of C st c in X /\ (dom f1) holds
f1 . c = r1 by PARTFUN2:76;
consider r2 being Real such that
A3: for c being Element of C st c in Y /\ (dom f2) holds
f2 . c = r2 by A1, PARTFUN2:76;

now

let c be Element of C; :: thesis: ( c in (X /\ Y) /\ (dom (f1 + f2)) implies (f1 + f2) . c = r1 + r2 )
assume c in (X /\ Y) /\ (dom (f1 + f2)) ; :: thesis: (f1 + f2) . c = r1 + r2
then A4: ( c in X /\ Y & c in dom (f1 + f2) ) by XBOOLE_0:def 4;
then ( c in X & c in Y & c in (dom f1) /\ (dom f2) ) by VALUED_1:def 1, XBOOLE_0:def 4;
then ( c in X & c in Y & c in dom f1 & c in dom f2 ) by XBOOLE_0:def 4;
then A5: ( c in X /\ (dom f1) & c in Y /\ (dom f2) ) by XBOOLE_0:def 4;
thus (f1 + f2) . c = (f1 . c) + (f2 . c) by A4, VALUED_1:def 1
.= r1 + (f2 . c) by A2, A5
.= r1 + r2 by A3, A5 ; :: thesis: verum

end;

hence (f1 + f2) | (X /\ Y) is constant by PARTFUN2:76; :: thesis: ( (f1 - f2) | (X /\ Y) is constant & (f1 (#) f2) | (X /\ Y) is constant )

now

let c be Element of C; :: thesis: ( c in (X /\ Y) /\ (dom (f1 - f2)) implies (f1 - f2) . c = r1 - r2 )
assume c in (X /\ Y) /\ (dom (f1 - f2)) ; :: thesis: (f1 - f2) . c = r1 - r2
then A6: ( c in X /\ Y & c in dom (f1 - f2) ) by XBOOLE_0:def 4;
then ( c in X & c in Y & c in (dom f1) /\ (dom f2) ) by VALUED_1:12, XBOOLE_0:def 4;
then ( c in X & c in Y & 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 XBOOLE_0:def 4;
thus (f1 - f2) . c = (f1 . c) - (f2 . c) by A6, VALUED_1:13
.= r1 - (f2 . c) by A2, A7
.= r1 - r2 by A3, A7 ; :: thesis: verum

end;

hence (f1 - f2) | (X /\ Y) is constant by PARTFUN2:76; :: thesis: (f1 (#) f2) | (X /\ Y) is constant

now

let c be Element of C; :: thesis: ( c in (X /\ Y) /\ (dom (f1 (#) f2)) implies (f1 (#) f2) . c = r1 * r2 )
assume c in (X /\ Y) /\ (dom (f1 (#) f2)) ; :: thesis: (f1 (#) f2) . c = r1 * r2
then ( c in X /\ Y & c in dom (f1 (#) f2) ) by XBOOLE_0:def 4;
then ( c in X & c in Y & c in (dom f1) /\ (dom f2) ) by VALUED_1:def 4, XBOOLE_0:def 4;
then ( c in X & c in Y & c in dom f1 & c in dom f2 ) by XBOOLE_0:def 4;
then A8: ( c in X /\ (dom f1) & c in Y /\ (dom f2) ) by XBOOLE_0:def 4;
thus (f1 (#) f2) . c = (f1 . c) * (f2 . c) by VALUED_1:5
.= r1 * (f2 . c) by A2, A8
.= r1 * r2 by A3, A8 ; :: thesis: verum

end;

hence (f1 (#) f2) | (X /\ Y) is constant by PARTFUN2:76; :: thesis: verum