 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 Th38:
  for S be non empty compact TopSpace,T be NormedLinearTopSpace
  for x1,x2 be Point of R_NormSpace_of_ContinuousFunctions(S,T),
  y1,y2 be Point of R_NormSpace_of_BoundedFunctions(the carrier of S,T)
  st x1=y1 & x2=y2 holds x1+x2=y1+y2
proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  let x1,x2 be Point of R_NormSpace_of_ContinuousFunctions(S,T),
      y1,y2 be Point of R_NormSpace_of_BoundedFunctions(the carrier of S,T);
  assume A1: x1=y1 & x2=y2;
  thus x1+x2 = ((the addF of RealVectSpace(the carrier of S,T))
         ||(ContinuousFunctions(S,T))).([x1,x2]) by RSSPACE:def 8,Th5
     .= (the addF of RealVectSpace(the carrier of S,T)).([x1,x2]) by FUNCT_1:49
     .= ((the addF of RealVectSpace(the carrier of S,T))
         ||(BoundedFunctions(the carrier of S,T))).([y1,y2]) by A1,FUNCT_1:49
     .=y1+y2 by RSSPACE:def 8,RSSPACE4:6;
end;
