
theorem
  for a be Real
  for X being compact non empty TopSpace
  for F,G being Point of R_Normed_Algebra_of_ContinuousFunctions(X)
  for f,g being RealMap of X holds
  (f=F & g=G implies ( G = a*F iff for x be Element of X holds g.x = a*f.x ))
proof
  let a be Real;
  let X be compact non empty TopSpace;
  let F,G be Point of R_Normed_Algebra_of_ContinuousFunctions(X);
  let f,g be RealMap of X;
  reconsider f1=F, g1=G as VECTOR of R_Algebra_of_ContinuousFunctions(X);
  G=a*F iff g1=a*f1;
  hence thesis by Th4;
end;
