
theorem Th28:
  for X being non empty compact TopSpace
  for F, G being Point of C_Normed_Algebra_of_ContinuousFunctions X holds
  ||.(F + G).|| <= ||.F.|| + ||.G.||
proof
  let X be non empty compact TopSpace;
  let F, G be Point of C_Normed_Algebra_of_ContinuousFunctions X;
  reconsider F1 = F, G1 = G
    as Point of C_Normed_Algebra_of_BoundedFunctions the carrier of X by Lm1;
A1:||.F.|| = ||.F1.|| by FUNCT_1:49;
A2:||.G.|| = ||.G1.|| by FUNCT_1:49;
A3:||.(F + G).|| = ||.(F1 + G1).|| by Lm4,Lm3;
  thus ||.(F + G).|| <= ||.F.|| + ||.G.|| by A1,A2,A3,CC0SP1:25;
end;
