
theorem
  for V, W being non empty ModuleStr over INT.Ring,
  a, b being Element of F_Real, f being FrForm of V,W holds
  (a+b)*f = a*f+b*f
  proof
    let V, W be non empty ModuleStr over INT.Ring,
    r, s be Element of F_Real, f be FrForm of V,W;
    now
      let v be Vector of V, w be Vector of W;
      thus ((r+s)*f).(v,w) = (r+s) * f.(v,w) by Def3
      .= r*f.(v,w) + s*f.(v,w)
      .= (r*f).(v,w) + s*f.(v,w) by Def3
      .= (r*f).(v,w) + (s*f).(v,w) by Def3
      .= (r*f + s*f).(v,w) by Def2;
    end;
    hence thesis;
  end;
