let M be non empty set ; :: thesis:
let V be ComplexNormSpace; :: thesis:
let f be PartFunc of M,the carrier of V; :: thesis:
let X, Y be set ; :: thesis:
assume that
A1: f is_bounded_on X and
A2: f is_bounded_on Y ; :: thesis:
consider r1 being Real such that
A3: for c being Element of M st c in X /\ (dom f) holds
||.(f /. c).|| <= r1 by A1, Def7;
consider r2 being Real such that
A4: for c being Element of M st c in Y /\ (dom f) holds
||.(f /. c).|| <= r2 by A2, Def7;
take r = (abs r1) + (abs r2); :: according to VFUNCT_2:def 7 :: thesis:
let c be Element of M; :: thesis:
assume A5: c in (X \/ Y) /\ (dom f) ; :: thesis:
then A6: c in dom f by XBOOLE_0:def 4;
A7: c in X \/ Y by A5, XBOOLE_0:def 4;

now

per cases by A7, XBOOLE_0:def 3;

suppose c in X ; :: thesis:

then c in X /\ (dom f) by A6, XBOOLE_0:def 4;
then A8: ||.(f /. c).|| <= r1 by A3;
A9: 0 <= abs r2 by COMPLEX1:132;
r1 <= abs r1 by ABSVALUE:11;
then ||.(f /. c).|| <= abs r1 by A8, XXREAL_0:2;
then ||.(f /. c).|| + 0 <= r by A9, XREAL_1:9;
hence ||.(f /. c).|| <= r ; :: thesis:

end;

suppose c in Y ; :: thesis:

then c in Y /\ (dom f) by A6, XBOOLE_0:def 4;
then A10: ||.(f /. c).|| <= r2 by A4;
A11: 0 <= abs r1 by COMPLEX1:132;
r2 <= abs r2 by ABSVALUE:11;
then ||.(f /. c).|| <= abs r2 by A10, XXREAL_0:2;
then 0 + ||.(f /. c).|| <= r by A11, XREAL_1:9;
hence ||.(f /. c).|| <= r ; :: thesis:

end;

end;

end;

hence ||.(f /. c).|| <= r ; :: thesis: