reserve x for set,
  K for Ring,
  r for Scalar of K,
  V for LeftMod of K,
  a,b,a1,a2 for Vector of V,
  A,A1,A2 for Subset of V,
  l for Linear_Combination of A,
  W for Subspace of V,
  Li for FinSequence of Submodules(V);

theorem Th10:
  a2 in a1 + W iff a1 - a2 in W
proof
  a1 - (a1 - a2) = a1 - a1 + a2 by RLVECT_1:29
    .= 0.V + a2 by VECTSP_1:19
    .= a2 by RLVECT_1:def 4;
  hence thesis by VECTSP_4:61;
end;
