 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 Th39:
  for S be non empty compact TopSpace,T be NormedLinearTopSpace
  for a be Real,
       x be Point of R_NormSpace_of_ContinuousFunctions(S,T),
  y be Point of R_NormSpace_of_BoundedFunctions(the carrier of S,T)
    st x=y holds a*x=a*y
proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  let a be Real,
      x be Point of R_NormSpace_of_ContinuousFunctions(S,T),
      y be Point of R_NormSpace_of_BoundedFunctions(the carrier of S,T);
  assume
A1: x=y;
   reconsider aa=a as Element of REAL by XREAL_0:def 1;
  thus a*x = ((the Mult of RealVectSpace(the carrier of S,T)) |
       [:REAL,(ContinuousFunctions(S,T)):]).([a,x])
      by RSSPACE:def 9,Th5
    .= (the Mult of RealVectSpace(the carrier of S,T)).([aa,x]) by FUNCT_1:49
    .= ((the Mult of RealVectSpace(the carrier of S,T))
       |[:REAL,(BoundedFunctions(the carrier of S,T)):]).[aa,y]
         by A1,FUNCT_1:49
    .=a*y by RSSPACE:def 9,RSSPACE4:6;
end;
