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

theorem Th18:
  for X be non empty closed_interval Subset of REAL for Y be RealNormSpace,
    f,g be Point of R_NormSpace_of_ContinuousFunctions(X,Y),
    f1,g1 be Point of R_NormSpace_of_BoundedFunctions(X,Y)
      st f1=f & g1=g holds f+g = f1+g1
proof
  let X be non empty closed_interval Subset of REAL,Y be RealNormSpace,
      f,g be Point of R_NormSpace_of_ContinuousFunctions(X,Y),
    f1,g1 be Point of R_NormSpace_of_BoundedFunctions(X,Y);
  assume A1: f1=f & g1=g;
  reconsider f2=f,g2=g as Point of R_VectorSpace_of_ContinuousFunctions(X,Y);
  reconsider f3=f2,g3=g2 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 f+g = f2+g2
          .= f3+g3 by A2,RLSUB_1:13
          .= f1 + g1 by A1;
end;
