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;
reserve R for Abelian add-associative right_zeroed right_complementable
  associative well-unital right_unital distributive non empty doubleLoopStr,
  F for non degenerated almost_left_invertible Ring,
  x for Scalar of F,
  V for add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital
   non empty
  ModuleStr over F,
  v for Vector of V;
reserve V for add-associative right_zeroed right_complementable RightMod-like
  non empty RightModStr over R;
reserve x for Scalar of R;
reserve v,w for Vector of V;

theorem Th33:
  -v*x = v*(-x) & w - v*x = w + v*(-x)
proof
A1: -v*x = (v*x) * (-1_R) by Th32
    .= v*(x* (-1_R)) by Def8
    .= v*(-(x* 1_R)) by VECTSP_1:8;
  hence -v* x = v*(-x);
  thus thesis by A1;
end;
