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 Th44:
  f is_bounded_on Y implies z(#)f is_bounded_on Y
proof
  assume f is_bounded_on Y;
  then consider r1 be Real such that
A1: for c be Element of M st c in Y /\ dom f holds ||.f/.c.|| <= r1;
   reconsider p = |.z.|*|.r1.| as Real;
  take p;
  let c be Element of M;
  assume
A2: c in Y /\ dom (z(#)f);
  then
A3: c in Y by XBOOLE_0:def 4;
A4: c in dom (z(#)f) by A2,XBOOLE_0:def 4;
  then c in dom f by Def2;
  then c in Y /\ dom f by A3,XBOOLE_0:def 4;
  then
A5: ||.f/.c.|| <= r1 by A1;
  r1 <= |.r1.| by ABSVALUE:4;
  then |.z.| >= 0 & ||.f/.c.|| <= |.r1.| by A5,COMPLEX1:46,XXREAL_0:2;
  then |.z.| * ||.(f/.c).|| <= |.z.|*|.r1.| by XREAL_1:64;
  then ||.z * (f/.c).|| <= p by CLVECT_1:def 13;
  hence thesis by A4,Def2;
end;
