
theorem Th5:
  for V being RealNormSpace,
      x,y be Point of V,
      z be Point of DualSp V holds
    (x + y) .|. z = x .|. z + y .|. z
proof
  let V be RealNormSpace,
      x,y be Point of V,
      z be Point of DualSp V;
  reconsider z1= z as linear-Functional of V by DUALSP01:def 10;
  thus (x + y) .|. z =z1.(x+y)
    .= x .|. z + y .|. z by HAHNBAN:def 2;
end;
