
theorem Th4:
  for X being non empty TopSpace
  for F,G,H being VECTOR of R_Algebra_of_ContinuousFunctions(X)
  for f,g,h being RealMap of X
  for a being Real
    holds
  (f=F & g=G implies ( G = a*F iff for x be Element of X holds g.x = a*f.x ))
proof
  let X be non empty TopSpace;
  let F,G,H be VECTOR of R_Algebra_of_ContinuousFunctions(X);
  let f,g,h be RealMap of X;
  let a be Real;
  assume
A1: f=F & g=G;
A2:R_Algebra_of_ContinuousFunctions(X) is
     Subalgebra of RAlgebra the carrier of X by C0SP1:6;
   reconsider f1=F, g1=G as VECTOR of
   RAlgebra 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,C0SP1:8;
     hence g.x=a*f.x by A1,FUNCSDOM:4;
   end;
   assume for x be Element of the carrier of X holds g.x=a*f.x;
   then g1=a*f1 by A1,FUNCSDOM:4;
   hence G =a*F by A2,C0SP1:8;
end;
