
theorem Th34:
  for X being non empty TopSpace
  for a be Complex,f be Function of the carrier of X,COMPLEX
                      holds support(a(#)f) c= support(f)
proof
  let X be non empty TopSpace;
  let a be Complex,f be Function of the carrier of X,COMPLEX;
  set CX= the carrier of X;
  reconsider h=a(#)f as Function of the carrier of X,COMPLEX;
  let x be object;
  assume x in support(a(#)f);
  then (a(#)f).x <>0 by PRE_POLY:def 7;
  then a*f.x <> 0 by VALUED_1:6;
  then f.x <> 0;
  hence x in support(f) by PRE_POLY:def 7;
end;
