reserve FS for non empty doubleLoopStr;
reserve F for Field;
reserve R for Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr,
  x, y, z for Scalar of R;
reserve SF for Skew-Field,
  x, y, z for Scalar of SF;
reserve R, R1, R2 for Ring;

theorem
  for K be add-associative right_zeroed right_complementable
right-distributive right_unital non empty doubleLoopStr for a be Element of K
  holds a * (- 1.K) = - a
proof
  let K be add-associative right_zeroed right_complementable
  right-distributive right_unital non empty doubleLoopStr;
  let x be Element of K;
  thus x * (- 1.K) = x * (0.K - 1.K) by RLVECT_1:14
    .= x * 0.K - x * 1.K by VECTSP_1:11
    .= 0.K - x * 1.K
    .= - x * 1.K by RLVECT_1:14
    .= - x;
end;
