 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 Th47:
  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;
  -x2=(-1)*x2 by RLVECT_1:16
    .=(-1)*y2 by A1,Th39
    .= -y2 by RLVECT_1:16;
  hence x1-x2 =y1-y2 by A1,Th38;
end;
