
theorem Th18:
  for a being Real
  for X being compact non empty TopSpace
  for F,G being Point of R_Normed_Algebra_of_ContinuousFunctions(X) holds
( ||.F.|| = 0 iff F = 0.R_Normed_Algebra_of_ContinuousFunctions(X)) &
(||.a*F.|| = |.a.| * ||.F.|| & ||.F+G.|| <= ||.F.|| + ||.G.||)
proof
  let a be Real;
  let X be compact non empty TopSpace;
  let F,G be Point of R_Normed_Algebra_of_ContinuousFunctions(X);
  reconsider F1=F, G1=G as Point of
    R_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;
  ||.F1.|| = 0 iff
    F1=0.R_Normed_Algebra_of_BoundedFunctions the carrier of X by C0SP1:32;
  hence ||.F.|| = 0 iff F = 0.R_Normed_Algebra_of_ContinuousFunctions(X)
                                        by Lm7,FUNCT_1:49;
  thus ||.a*F.|| = ||.a*F1.|| by Lm5,Lm3
                .=|.a.| * ||.F1.|| by C0SP1:32
                .=|.a.| * ||.F.|| by FUNCT_1:49;
   thus ||.F+G.|| <= ||.F.|| + ||.G.|| by A1,A2,A3,C0SP1:32;
end;
