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 holds a * Sum(<*>(the carrier of V)) =
  0.V
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;
  thus a * Sum(<*>(the carrier of V)) = a * 0.V by RLVECT_1:43
    .= 0.V by VECTSP_1:14;
end;
