
theorem Th11:
  for X being compact non empty TopSpace
  for F being Point of R_Normed_Algebra_of_ContinuousFunctions(X) holds
  (Mult_(ContinuousFunctions(X), RAlgebra the carrier of X)).(1,F) = F
proof
  let X be compact non empty TopSpace;
  let F be Point of R_Normed_Algebra_of_ContinuousFunctions(X);
  set X1 = ContinuousFunctions(X);
  reconsider f1 = F as Element of X1;
A1:[jj,f1] in [:REAL,X1:];
  thus (Mult_(ContinuousFunctions(X), RAlgebra the carrier of X)).(1,F)
    = ((the Mult of RAlgebra the carrier of X)| [:REAL,X1:]).(1,f1)
       by C0SP1:def 11
   .= (the Mult of RAlgebra the carrier of X).(1,f1) by A1,FUNCT_1:49
   .= F by FUNCSDOM:12;
end;
