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

theorem Th15:
  for X be non empty closed_interval Subset of REAL,
      Y be RealNormSpace, f,g,h be Point of
         R_NormSpace_of_ContinuousFunctions(X,Y),
      f9,g9,h9 be continuous PartFunc of REAL,Y
  st f9=f & g9=g & h9=h & dom f9=X & dom g9=X & dom h9=X
    holds (h = f+g iff for x be Element of X
  holds h9/.x = f9/.x + g9/.x)
proof
  let X be non empty closed_interval Subset of REAL,
      Y be RealNormSpace,
      f,g,h be Point of R_NormSpace_of_ContinuousFunctions(X,Y);
  reconsider f1=f, g1=g, h1=h
      as VECTOR of R_VectorSpace_of_ContinuousFunctions(X,Y);
A1: h=f+g iff h1=f1 + g1;
  let f9,g9,h9 be continuous PartFunc of REAL,Y;
  thus thesis by A1,Th10;
end;
