
theorem Th25:
  for X being non empty compact TopSpace holds
  ||.0.C_Normed_Algebra_of_ContinuousFunctions X.|| = 0
proof
  let X be non empty compact TopSpace;
  set F = 0.C_Normed_Algebra_of_ContinuousFunctions X;
  reconsider F1 = F
    as Point of C_Normed_Algebra_of_BoundedFunctions the carrier of X by Lm1;
  (||.F1.|| = 0 iff F1
     = 0.(C_Normed_Algebra_of_BoundedFunctions the carrier of X)) by CC0SP1:25;
  hence thesis by Lm7,FUNCT_1:49;
end;
