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 Th45:
  f is_bounded_on Y implies ||.f.|||Y is bounded & -f is_bounded_on Y
proof
  assume
A1: f is_bounded_on Y;
  then consider r1 be Real such that
A2: for c be Element of M st c in Y /\ dom f holds ||.f/.c.|| <= r1;
  now
    let c be object;
    assume
A3: c in Y /\ dom (||.f.||);
    then
A4: c in Y by XBOOLE_0:def 4;
A5: c in dom (||.f.||) by A3,XBOOLE_0:def 4;
    then c in dom f by NORMSP_0:def 3;
    then c in Y /\ dom f by A4,XBOOLE_0:def 4;
    then ||.f/.c.|| >= 0 & ||.f/.c.|| <= r1 by A2,CLVECT_1:105;
    then |.||.f/.c.||.| <= r1 by ABSVALUE:def 1;
    hence |.(||.f.||).c.| <= r1 by A5,NORMSP_0:def 3;
  end;
  hence ||.f.|||Y is bounded by RFUNCT_1:73;
  (-1r)(#)f is_bounded_on Y by A1,Th44;
  hence thesis by Th23;
end;
