reserve V for RealLinearSpace,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n for Element of NAT,
  v for VECTOR of V,
  KL1, KL2 for Linear_Combination of V,
  X for Subset of V;

theorem Th2:
  for V being RealLinearSpace, A being Subset of V st A is
  linearly-independent holds ex I being Basis of V st A c= I
proof
  let V be RealLinearSpace, A be Subset of V;
  assume A is linearly-independent;
  then consider B being Subset of V such that
A1: A c= B and
A2: B is linearly-independent & Lin(B) = the RLSStruct of V by RLVECT_3:24;
  reconsider B as Basis of V by A2,RLVECT_3:def 3;
  take B;
  thus thesis by A1;
end;
