
theorem
  for K be left-distributive non empty doubleLoopStr for V,W be non
empty ModuleStr over K for a,b be Element of K for f be Form of V,W holds (a+b)
  *f = a*f+b*f
proof
  let K be left-distributive non empty doubleLoopStr, V,W be non empty
  ModuleStr over K, r,s be Element of K, 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 Def3
      .= r*f.(v,w) + s*f.(v,w) by VECTSP_1:def 3
      .= (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;
