
theorem
  for V being RealNormSpace,
      y,z be Point of V,
      x be Point of DualSp V,
      a,b be Real holds
    (a*y + b*z) .|. x = a * y .|. x + b * z .|. x
proof
  let V be RealNormSpace,
      y,z be Point of V,
      x be Point of DualSp V,
      a,b be Real;
  thus (a*y + b*z) .|. x = (a*y) .|. x + (b*z) .|. x by Th5
  .=a * y .|. x + (b*z) .|. x by Th6
  .=a * y .|. x + b * z .|. x by Th6;
end;
