 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 Th64:
  for X be non empty TopSpace,T be NormedLinearTopSpace
  for x1,x2 be Point of R_Normed_Space_of_C_0_Functions (X,T),
  y1,y2 be Point of R_NormSpace_of_BoundedFunctions(the carrier of X,T)
                    st x1=y1 & x2=y2 holds x1+x2=y1+y2
proof
  let X be non empty TopSpace,T be NormedLinearTopSpace;
  let x1,x2 be Point of R_Normed_Space_of_C_0_Functions (X,T),
  y1,y2 be Point of R_NormSpace_of_BoundedFunctions(the carrier of X,T);
  assume
A1: x1=y1 & x2=y2;
  thus x1+x2 = ((the addF of RealVectSpace(the carrier of X,T))
            ||C_0_Functions(X,T)).([x1,x2]) by RSSPACE:def 8
      .= (the addF of RealVectSpace(the carrier of X,T)).([x1,x2])
        by FUNCT_1:49
      .= ((the addF of RealVectSpace(the carrier of X,T))
          || (BoundedFunctions(the carrier of X,T)).([y1,y2])) by A1,FUNCT_1:49
          .=y1+y2 by RSSPACE:def 8,RSSPACE4:6;
end;
