
theorem Th19:
  for X be non empty set for Y be RealNormSpace for f,g,h be Point
of R_NormSpace_of_BoundedFunctions(X,Y) for f9,g9,h9 be bounded Function of X,
the carrier of Y st f9=f & g9=g & h9=h holds (h = f+g iff for x be Element of X
  holds h9.x = f9.x + g9.x )
proof
  let X be non empty set;
  let Y be RealNormSpace;
  let f,g,h be Point of R_NormSpace_of_BoundedFunctions(X,Y);
  reconsider f1=f, g1=g, h1=h as VECTOR of R_VectorSpace_of_BoundedFunctions(X
  ,Y);
A1: h=f+g iff h1=f1+g1;
  let f9,g9,h9 be bounded Function of X,the carrier of Y;
  assume f9=f & g9=g & h9=h;
  hence thesis by A1,Th8;
end;
