theorem
  for R being add-associative right_zeroed right_complementable Abelian
associative well-unital distributive non empty doubleLoopStr, a being Element
  of R for V being Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over R, v,u,w being Element of V holds a *
  Sum<* v,u,w *> = a * v + a * u + a * w
proof
  let R be add-associative right_zeroed right_complementable Abelian
associative well-unital distributive non empty doubleLoopStr, a be Element of
  R;
  let V be Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over R, v,u,w be Element of V;
  thus a * Sum<* v,u,w *> = a * (v + u + w) by RLVECT_1:46
    .= a * (v + u) + a * w by VECTSP_1:def 14
    .= a * v + a * u + a * w by VECTSP_1:def 14;
end;
