 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th55:
  for X be non empty TopSpace,T be RealLinearSpace,
      f be Function of X,T,
      a be Real holds
    support(a(#)f) c= support(f)
proof
  let X be non empty TopSpace,T be RealLinearSpace;
  let f be Function of X,T,
      a be Real;
  set CX = the carrier of X;
  reconsider h=a(#)f as Function of X,T;
  now let x be object;
    assume x in support(a(#)f); then
A1: x in dom (a(#)f) & (a(#)f)/.x <>0.T by Def10; then
    a*f/.x <> 0.T by VFUNCT_1:def 4; then
A3: f/.x <> 0.T;
    x in dom f by A1,VFUNCT_1:def 4;
    hence x in support(f) by A3,Def10;
  end;
  hence thesis;
end;
