
theorem
  for X being non empty compact TopSpace
  for F being Point of C_Normed_Algebra_of_ContinuousFunctions X holds
    (Mult_(CContinuousFunctions X,CAlgebra the carrier of X)).(1r,F) = F
proof
  let X be non empty compact TopSpace;
  let F be Point of C_Normed_Algebra_of_ContinuousFunctions X;
  set X1 = CContinuousFunctions X;
  reconsider f1 = F as Element of CContinuousFunctions X;
A1:[1,f1] in [:COMPLEX,(CContinuousFunctions X):] by ZFMISC_1:87;
  (Mult_ ((CContinuousFunctions X),(CAlgebra the carrier of X))).(1,F)
     = (( the Mult of CAlgebra the carrier of X) |
                 [:COMPLEX,(CContinuousFunctions X):]).(1,f1) by CC0SP1:def 3
    .= (the Mult of CAlgebra the carrier of X).(1,f1) by A1,FUNCT_1:49
    .= F by CFUNCDOM:12;
  hence thesis;
end;
