let X, Y be set ; :: thesis: for C being non empty set
for f1, f2 being PartFunc of C,COMPLEX st f1 | X is bounded & f2 | Y is bounded holds
( (f1 (#) f2) | (X /\ Y) is bounded & (f1 - f2) | (X /\ Y) is bounded )
let C be non empty set ; :: thesis: for f1, f2 being PartFunc of C,COMPLEX st f1 | X is bounded & f2 | Y is bounded holds
( (f1 (#) f2) | (X /\ Y) is bounded & (f1 - f2) | (X /\ Y) is bounded )
let f1, f2 be PartFunc of C,COMPLEX; :: thesis: ( f1 | X is bounded & f2 | Y is bounded implies ( (f1 (#) f2) | (X /\ Y) is bounded & (f1 - f2) | (X /\ Y) is bounded ) )
assume that
A1: f1 | X is bounded and
A2: f2 | Y is bounded ; :: thesis: ( (f1 (#) f2) | (X /\ Y) is bounded & (f1 - f2) | (X /\ Y) is bounded )
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, Th68;
consider r2 being Real such that
A4: for c being Element of C st c in Y /\ (dom f2) holds
|.(f2 /. c).| <= r2 by A2, Th68;

now :: thesis: ex r being set st
for c being Element of C st c in (X /\ Y) /\ (dom (f1 (#) f2)) holds
|.((f1 (#) f2) /. c).| <= r

take r = r1 * r2; :: thesis: for c being Element of C st c in (X /\ Y) /\ (dom (f1 (#) f2)) holds
|.((f1 (#) f2) /. c).| <= r
let c be Element of C; :: thesis: ( c in (X /\ Y) /\ (dom (f1 (#) f2)) implies |.((f1 (#) f2) /. c).| <= r )
assume A5: c in (X /\ Y) /\ (dom (f1 (#) f2)) ; :: thesis: |.((f1 (#) f2) /. c).| <= r
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 Th3;
then c in dom f1 by XBOOLE_0:def 4;
then c in X /\ (dom f1) by A7, XBOOLE_0:def 4;
then A10: |.(f1 /. c).| <= r1 by A3;
A11: c in Y by A6, XBOOLE_0:def 4;
c in dom f2 by A9, XBOOLE_0:def 4;
then c in Y /\ (dom f2) by A11, XBOOLE_0:def 4;
then A12: |.(f2 /. c).| <= r2 by A4;
( 0 <= |.(f1 /. c).| & 0 <= |.(f2 /. c).| ) by COMPLEX1:46;
then |.(f1 /. c).| * |.(f2 /. c).| <= r by A10, A12, XREAL_1:66;
then |.((f1 /. c) * (f2 /. c)).| <= r by COMPLEX1:65;
hence |.((f1 (#) f2) /. c).| <= r by A8, Th3; :: thesis: verum

end;

hence (f1 (#) f2) | (X /\ Y) is bounded by Th68; :: thesis: (f1 - f2) | (X /\ Y) is bounded
(- f2) | Y is bounded by A2, Th73;
hence (f1 - f2) | (X /\ Y) is bounded by A1, Th74; :: thesis: verum