reserve x,y for object,
  R for Ring,
  V for LeftMod of R,
  L for Linear_Combination of V,
  a for Scalar of R,
  v,u for Vector of V,
  F,G for FinSequence of the carrier of V,
  C for finite Subset of V;
reserve X,Y,Z for set,
  A,B for Subset of V,
  T for finite Subset of V,
  l for Linear_Combination of A,
  f,g for Function of the carrier of V,the carrier of R;

theorem Th14:
  (0).V is free
proof
  set W = (0).V;
  reconsider B9 = {}(the carrier of V) as Subset of W by SUBSET_1:1;
  reconsider V9 = V as Subspace of V by VECTSP_4:24;
A1: B9 = {}(the carrier of W); then
A2: B9 is linearly-independent;
  (0).V9 = (0).W by VECTSP_4:37;
  then Lin(B9) = W by A1,Th6;
  then B9 is base by A2;
  hence thesis;
end;
