reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;

theorem LmRankSX11:
  for V being finite-rank free Z_Module,
  A being linearly-independent Subset of V holds
  ex I being finite linearly-independent Subset of V,
     a being Element of INT.Ring
  st a <> 0.INT.Ring & A c= I & a (*) V is Submodule of Lin(I)
  proof
    let V be finite-rank free Z_Module,
    A be linearly-independent Subset of V;
    consider I be finite Subset of V, a be Element of INT.Ring such that
    P0: a <> 0.INT.Ring and
    P1: A c= I and
    P2: I is linearly-independent and
    P3: for v being VECTOR of V holds a * v in Lin(I) by LmTF1A;
    reconsider I as finite linearly-independent Subset of V by P2;
    take I, a;
    thus a <> 0.INT.Ring & A c= I by P0,P1;
    for v being VECTOR of V st v in a (*) V holds v in Lin(I)
    proof
      let v be VECTOR of V;
      assume v in a (*) V;
      then consider w be VECTOR of V such that
      P4: v = a * w;
      thus v in Lin(I) by P3,P4;
    end;
    hence a (*) V is Submodule of Lin(I) by ZMODUL01:44;
  end;
