
theorem Th23:
  for X being non empty set,
      a being Complex,
      f, g being Function of X,COMPLEX,
      F, G being Point of C_Normed_Algebra_of_BoundedFunctions(X)
                               st f = F & g = G holds
  ( G = a*F iff for x being Element of X holds g.x = a*(f.x))
proof
  let X be non empty set,
      a be Complex,
      f, g be Function of X,COMPLEX,
      F, G be Point of C_Normed_Algebra_of_BoundedFunctions(X);
  reconsider f1 = F, g1 = G as VECTOR of C_Algebra_of_BoundedFunctions(X);
A1: G = a * F iff g1 = a * f1;
  assume f = F & g = G;
  hence thesis by A1,Th6;
end;
