 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th46:
  for a being Real
  for S be non empty compact TopSpace,T be NormedLinearTopSpace
  for F,G being Point of R_NormSpace_of_ContinuousFunctions(S,T) holds
( ||.F.|| = 0 iff F = 0.R_NormSpace_of_ContinuousFunctions(S,T)) &
(||.a*F.|| = |.a.| * ||.F.|| & ||.F+G.|| <= ||.F.|| + ||.G.||)
proof
  let a be Real;
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  let F,G be Point of R_NormSpace_of_ContinuousFunctions(S,T);
  reconsider F1=F, G1=G as Point of
    R_NormSpace_of_BoundedFunctions(the carrier of S,T) by Th34;
A1:||.F.|| = ||.F1.|| by FUNCT_1:49;
A2:||.G.|| = ||.G1.|| by FUNCT_1:49;
A3:||.F+G.|| = ||.F1+G1.|| by Th38,Th36;
  ||.F1.|| = 0 iff
    F1=0.R_NormSpace_of_BoundedFunctions(the carrier of S,T)
      by NORMSP_0:def 5;
  hence ||.F.|| = 0 iff F = 0.R_NormSpace_of_ContinuousFunctions(S,T)
                                        by FUNCT_1:49,Th41;
  thus ||.a*F.|| = ||.a*F1.|| by Th39,Th36
                .=|.a.| * ||.F1.|| by NORMSP_1:def 1
                .=|.a.| * ||.F.|| by FUNCT_1:49;
   thus ||.F+G.|| <= ||.F.|| + ||.G.|| by A1,A2,A3,NORMSP_1:def 1;
end;
