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 Th4:
  V is Subspace of (Omega).V
proof
  set W=(Omega).V;
A1: W = the ModuleStr of V by VECTSP_4:def 4;
  then
A2: 0.V = 0.W;
  dom(the lmult of W) = [:the carrier of K, the carrier of W:] by FUNCT_2:def 1
;
  then
A3: the lmult of V = (the lmult of W) | [:the carrier of K, the carrier of V
  :] by A1,RELAT_1:68;
  dom(the addF of W) = [:the carrier of W,the carrier of W:] by FUNCT_2:def 1;
  then the addF of V = (the addF of W)||the carrier of V by A1,RELAT_1:68;
  hence thesis by A1,A2,A3,VECTSP_4:def 2;
end;
