 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 Th66:
  for a be Real
  for X be non empty TopSpace,T be NormedLinearTopSpace
  for F,G be Point of R_Normed_Space_of_C_0_Functions (X,T) holds
  (||.F.|| = 0 iff F = 0.R_Normed_Space_of_C_0_Functions (X,T) ) &
  ||.a*F.|| = |.a.| * ||.F.|| & ||.F+G.|| <= ||.F.|| + ||.G.||
proof
  let a be Real;
  let X be non empty TopSpace,T be NormedLinearTopSpace;
  let F,G be Point of R_Normed_Space_of_C_0_Functions (X,T);
A1:||.F.|| = 0 iff F = 0.R_Normed_Space_of_C_0_Functions (X,T)
  proof
    reconsider FB=F as
       Point of R_NormSpace_of_BoundedFunctions(the carrier of X,T) by Th60;
A2: 0.R_Normed_Space_of_C_0_Functions (X,T) =X-->0.T by Th63;
    ||.FB.|| = 0 iff
          FB = 0.R_NormSpace_of_BoundedFunctions(the carrier of X,T)
                                                    by RSSPACE4:21;
    hence thesis by A2,RSSPACE4:15,FUNCT_1:49;
  end;
A3:||.a*F.|| = |.a.| * ||.F.||
  proof
    reconsider FB=F as
       Point of R_NormSpace_of_BoundedFunctions(the carrier of X,T) by Th60;
A4: ||.FB.||=||.F.|| by FUNCT_1:49;
A5: a*FB=a*F by Th65;
    reconsider aFB=a*FB as Point of R_NormSpace_of_BoundedFunctions
                                (the carrier of X,T);
    reconsider aF=a*F as Point of R_Normed_Space_of_C_0_Functions (X,T);
    ||.aFB.||=||.aF.|| by A5,FUNCT_1:49;
    hence thesis by A4,RSSPACE4:21;
  end;
  ||.F+G.|| <= ||.F.|| + ||.G.||
  proof
    reconsider FB=F,GB=G as
       Point of R_NormSpace_of_BoundedFunctions(the carrier of X,T) by Th60;
A6: ||.FB.||=||.F.|| & ||.GB.||=||.G.|| by FUNCT_1:49;
    FB+GB=F+G by Th64; then
    ||.FB+GB.||=||.F+G.|| by FUNCT_1:49;
    hence thesis by A6,RSSPACE4:21;
  end;
  hence thesis by A1,A3;
end;
