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