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;
