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 Th47:
  vector(LC_Z_Module(V),L1) + vector(LC_Z_Module(V),L2) = L1 + L2
  proof
    set v2 = vector(LC_Z_Module(V),L2);
    A1: L1 = @@L1 & L2 = @@L2;
    L2 in the carrier of LC_Z_Module(V) by Def29;
    then
    A2: L2 in LC_Z_Module(V);
    L1 in the carrier of LC_Z_Module(V) by Def29;
    then L1 in LC_Z_Module(V);
    hence vector(LC_Z_Module(V),L1)
    + vector(LC_Z_Module(V),L2) = LCAdd(V). [L1,v2] by RLVECT_2:def 1
    .= LCAdd(V).(@L1,@L2) by A2,RLVECT_2:def 1
    .= L1 + L2 by A1,Def32;
  end;
