
theorem Th6:
  for X being non empty set, a being Complex,
      F, G being VECTOR of C_Algebra_of_BoundedFunctions(X),
      f, g being Function of X,COMPLEX st f=F & g=G holds
  ( G = a*F iff for x be Element of X holds g.x = a*(f.x) )
proof
  let X be non empty set,
      a be Complex,
      F, G be VECTOR of C_Algebra_of_BoundedFunctions(X),
      f, g be Function of X,COMPLEX;
  assume
A1: f=F & g=G;
A2:C_Algebra_of_BoundedFunctions X is ComplexSubAlgebra of CAlgebra X by Th2;
  reconsider f1=F, g1=G as VECTOR of CAlgebra X by TARSKI:def 3;
  hereby assume
A3: G = a*F;
   let x be Element of X;
   g1=a*f1 by A2,A3,Th3;
   hence g.x=a*f.x by A1,CFUNCDOM:4;
  end;
  assume for x be Element of X holds g.x=a*f.x; then
  g1=a*f1 by A1,CFUNCDOM:4;
  hence G =a*F by A2,Th3;
end;
