
theorem
  for X being compact non empty TopSpace
  for F be Point of R_Normed_Algebra_of_ContinuousFunctions(X) holds
  F = 0.R_Normed_Algebra_of_ContinuousFunctions(X) implies 0 = ||.F.||
proof
  let X be compact non empty TopSpace;
  let F be Point of R_Normed_Algebra_of_ContinuousFunctions(X);
  assume
A1: F = 0.R_Normed_Algebra_of_ContinuousFunctions(X);
  reconsider F1=F as Point of
    R_Normed_Algebra_of_BoundedFunctions the carrier of X by Lm1;
A2:||.F.|| =||.F1.|| by FUNCT_1:49;
  F= X--> 0 by A1,Th12; then
  F1 = 0.R_Normed_Algebra_of_BoundedFunctions the carrier of X by C0SP1:25;
  hence thesis by A2,C0SP1:28;
end;
