reserve x for set,
  K for Ring,
  r for Scalar of K,
  V, M, M1, M2, N for LeftMod of K,
  a for Vector of V,
  m, m1, m2 for Vector of M,
  n, n1, n2 for Vector of N,
  A for Subset of V,
  l for Linear_Combination of A,
  W, W1, W2, W3 for Subspace of V;

theorem Th38:
  W1 c= W2 iff [#]W1 c= [#]W2
proof
  thus W1 c= W2 implies [#]W1 c= [#]W2 by Th36;
  assume [#]W1 c= [#]W2;
  then for a st a in W1 holds a in W2;
  hence thesis by Th37;
end;
