
theorem Th11:
  for X being non empty TopSpace
  for F, G being VECTOR of (C_Algebra_of_ContinuousFunctions X)
  for f, g being Function of the carrier of X,COMPLEX
  for a being Complex 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 TopSpace;
  let F, G be VECTOR of (C_Algebra_of_ContinuousFunctions X);
  let f, g be Function of the carrier of X,COMPLEX;
  let a be Complex;
  assume
A1: f = F & g = G;
A2:C_Algebra_of_ContinuousFunctions X
          is ComplexSubAlgebra of CAlgebra the carrier of X by CC0SP1:2;
  reconsider f1 = F, g1 = G
      as VECTOR of (CAlgebra the carrier of X) by TARSKI:def 3;
  hereby
    assume
A3: G = a * F;
    let x be Element of the carrier of X;
    g1 = a * f1 by A2,A3,CC0SP1:3;
    hence g . x = a * (f . x) by A1,CFUNCDOM:4;
  end;
  assume for x being Element of the carrier of X holds g . x = a * (f . x);
  then g1 = a * f1 by A1,CFUNCDOM:4;
  hence G = a * F by A2,CC0SP1:3;
end;
