theorem
  for V, W being non empty ModuleStr over INT.Ring,
      a, b being Element of INT.Ring,
  f being Form 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 INT.Ring,
      f be Form 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 BLDef3
      .= r*f.(v,w) + s*f.(v,w)
      .= (r*f).(v,w) + s*f.(v,w) by BLDef3
      .= (r*f).(v,w) + (s*f).(v,w) by BLDef3
      .= (r*f + s*f).(v,w) by BLDef2;
    end;
    hence thesis;
  end;
