
theorem Th20:
  for X be non empty set, Y be ComplexNormSpace, f,g,h be Point of
  C_NormSpace_of_BoundedFunctions(X,Y), 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 ComplexNormSpace;
  let f,g,h be Point of C_NormSpace_of_BoundedFunctions(X,Y);
  reconsider f1=f, g1=g, h1=h as VECTOR of C_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,Th9;
end;
