
theorem Th20:
  for X be non empty set for Y be RealNormSpace for f,h be Point
of R_NormSpace_of_BoundedFunctions(X,Y) for f9,h9 be bounded Function of X,the
carrier of Y st f9=f & h9=h
for a be Real holds h = a*f iff for x be Element of
  X holds h9.x = a*f9.x
proof
  let X be non empty set;
  let Y be RealNormSpace;
  let f,h be Point of R_NormSpace_of_BoundedFunctions(X,Y);
  let f9,h9 be bounded Function of X,the carrier of Y such that
A1: f9=f & h9=h;
  reconsider h1=h as VECTOR of R_VectorSpace_of_BoundedFunctions(X,Y);
  reconsider f1=f as VECTOR of R_VectorSpace_of_BoundedFunctions(X,Y);
  let a be Real;
  h=a*f iff h1=a*f1;
  hence thesis by A1,Th9;
end;
