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
  (v - w)*x = v*x - w*x
proof
  (v - w)*x = (v + (-w))*x .= v*x + (-w) *x by Def8
    .= v*x + (-w * x) by Th34;
  hence thesis;
end;
