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
  M c= N implies 0.M = 0.N & (m1 = n1 & m2 = n2 implies m1 + m2 = n1 +
n2) & (m = n implies r * m = r * n) & (m = n implies - n = - m) & (m1 = n1 & m2
  = n2 implies m1 - m2 = n1 - n2) & 0.N in M & 0.M in N & (n1 in M & n2 in M
implies n1 + n2 in M) & (n in M implies r * n in M) & (n in M implies - n in M)
  & (n1 in M & n2 in M implies n1 - n2 in M)

by VECTSP_4:11,VECTSP_4:13,VECTSP_4:14,VECTSP_4:15,VECTSP_4:16,VECTSP_4:17,
