reserve x, y, y1, y2 for set;
reserve R for Ring;
reserve V for LeftMod of R;
reserve u, v, w for VECTOR of V;
reserve F, G, H, I for FinSequence of V;
reserve i, j, k, n for Element of NAT;
reserve f, f9, g for sequence of V;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve a, b for Element of R;
reserve G, H1, H2, F, F1, F2, F3 for FinSequence of V;
reserve A, B for Subset of V,
  v1, v2, v3, u1, u2, u3 for Vector of V,
  f for Function of V, R,
  i for Element of NAT;
reserve l, l1, l2 for Linear_Combination of A;
 reserve e, e1, e2 for Element of LinComb(V);

theorem
  vector(LC_Z_Module(V),L1) - vector(LC_Z_Module(V),L2) = L1 - L2
  proof
    - L2 in LinComb(V) by Def29; then
A1: - L2 in LC_Z_Module(V);
    - vector(LC_Z_Module(V),L2) = - L2 by Th49
    .= vector(LC_Z_Module(V),- L2) by A1,RLVECT_2:def 1;
    hence thesis by Th47;
  end;
