 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 Th65:
  for X be non empty TopSpace,T be NormedLinearTopSpace
  for a be Real,x be Point of R_Normed_Space_of_C_0_Functions (X,T),
  y be Point of R_NormSpace_of_BoundedFunctions(the carrier of X,T)
                     st x=y holds a*x=a*y
proof
  let X be non empty TopSpace,T be NormedLinearTopSpace;
  let a be Real, x be Point of R_Normed_Space_of_C_0_Functions (X,T),
  y be Point of R_NormSpace_of_BoundedFunctions(the carrier of X,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 X,T))
  |[:REAL,(C_0_Functions(X,T)):]).([a,x]) by RSSPACE:def 9
   .= (the Mult of RealVectSpace(the carrier of X,T)).([aa,x]) by FUNCT_1:49
   .= ((the Mult of RealVectSpace(the carrier of X,T))
  |[:REAL,(BoundedFunctions(the carrier of X,T)) :]).([aa,y]) by A1,FUNCT_1:49
    .= a*y by RSSPACE:def 9,RSSPACE4:6;
end;
