theorem
  for R being non degenerated almost_left_invertible Ring
  for V being LeftMod of R for A being Subset of V st
  A is linearly-independent holds ex I being Basis of V st A c= I
proof
  let R be non degenerated almost_left_invertible Ring;
  let V be LeftMod of R;
  let 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 base by Th18;
  reconsider B as Basis of V by A2;
  take B;
  thus thesis by A1;
end;
