
theorem
  for K be right-distributive non empty doubleLoopStr for V,W be non
empty ModuleStr over K, a be Element of K for f,g be Form of V,W holds a*(f+g)
  = a*f+a*g
proof
  let K be right-distributive non empty doubleLoopStr, V,W be non empty
  ModuleStr over K, r be Element of K, f,g be Form 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) by VECTSP_1:def 2
      .= (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;
