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 being Element of V holds a * Sum
  <* v,u *> = a * v + a * u
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 be Element of V;
  thus a * Sum<* v,u *> = a * (v + u) by RLVECT_1:45
    .= a * v + a * u by VECTSP_1:def 14;
end;
