reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;
reserve X,Y for set;

theorem Th46:
  f1 is_bounded_on X & f2 is_bounded_on Y implies f1+f2 is_bounded_on (X /\ Y)
proof
  assume that
A1: f1 is_bounded_on X and
A2: f2 is_bounded_on Y;
  consider r1 be Real such that
A3: for c be Element of M st c in X /\ dom f1 holds ||.(f1/.c).|| <= r1
  by A1;
  consider r2 be Real such that
A4: for c be Element of M st c in Y /\ dom f2 holds ||.(f2/.c).|| <= r2
  by A2;
  take r=r1+r2;
  let c be Element of M;
  assume
A5: c in X /\ Y /\ dom (f1+f2);
  then
A6: c in X /\ Y by XBOOLE_0:def 4;
  then
A7: c in Y 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 VFUNCT_1:def 1;
  then c in dom f2 by XBOOLE_0:def 4;
  then c in Y /\ dom f2 by A7,XBOOLE_0:def 4;
  then
A10: ||.(f2/.c).|| <= r2 by A4;
A11: c in X by A6,XBOOLE_0:def 4;
  c in dom f1 by A9,XBOOLE_0:def 4;
  then c in X /\ dom f1 by A11,XBOOLE_0:def 4;
  then ||.(f1/.c).|| <= r1 by A3;
  then ||.(f1/.c) + (f2/.c).|| <= ||.(f1/.c).|| + ||.f2/.c.|| & ||.(f1/.c).||
  + ||. (f2/.c).|| <= r by A10,CLVECT_1:def 13,XREAL_1:7;
  then ||.(f1/.c) + (f2/.c).|| <= r by XXREAL_0:2;
  hence thesis by A8,VFUNCT_1:def 1;
end;
