reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem Th19:
  for X be non empty closed_interval Subset of REAL for Y be RealNormSpace,
      f be Point of R_NormSpace_of_ContinuousFunctions(X,Y),
     f1 be Point of R_NormSpace_of_BoundedFunctions(X,Y) st f1=f
      holds a*f = a*f1
proof
   let X be non empty closed_interval Subset of REAL,Y be RealNormSpace,
      f be Point of R_NormSpace_of_ContinuousFunctions(X,Y),
      f1 be Point of R_NormSpace_of_BoundedFunctions(X,Y);
   assume A1: f1=f;
   reconsider f2=f as Point of R_VectorSpace_of_ContinuousFunctions(X,Y);
   reconsider f3=f2 as Point of R_VectorSpace_of_BoundedFunctions(X,Y)
   by TARSKI:def 3;
A2: R_VectorSpace_of_ContinuousFunctions(X,Y)
    is Subspace of R_VectorSpace_of_BoundedFunctions(X,Y) by RSSPACE:11;
   thus a*f = a*f2
           .= a*f3 by A2,RLSUB_1:14
           .= a*f1 by A1;
end;
