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 Th48:
  a * vector(LC_Z_Module(V),L) = a * L
  proof
    A1: @@L = L;
    L in the carrier of LC_Z_Module(V) by Def29;
    then L in LC_Z_Module(V);
    hence a * vector(LC_Z_Module(V),L) = LCMult(V). [a,@L] by RLVECT_2:def 1
    .= a * L by A1,Def33;
  end;
