reserve V,W for Z_Module;

theorem
  for V being finite-rank free Z_Module holds
  rank V = 2 iff ex u, v being VECTOR of V st u <> v &
  {u, v} is linearly-independent & (Omega).V = Lin{u, v}
  proof
    let V be finite-rank free Z_Module;
    hereby
      consider I being finite Subset of V such that
      A1: I is Basis of V by ZMODUL03:def 3;
      assume rank V = 2; then
  A2: card I = 2 by A1,ZMODUL03:def 5;
      then consider u being object such that
  A3: u in I by CARD_1:27,XBOOLE_0:def 1;
      reconsider u as VECTOR of V by A3;
      now
        assume I c= {u};
        then card I <= card {u} by NAT_1:43;
        then 2 <= 1 by A2,CARD_1:30;
        hence contradiction;
      end;
      then consider v being object such that
      A4: v in I and
      A5: not v in {u};
      reconsider v as VECTOR of V by A4;
      A6: v <> u by A5,TARSKI:def 1;
      A7:
      now
        assume not I c= {u, v};
        then consider w being object such that
        A8: w in I and
        A9: not w in {u, v};
        {u, v, w} c= I by A3,A4,A8,ENUMSET1:def 1; then
        A10: card {u, v, w} <= card I by NAT_1:43;
        w <> u & w <> v by A9,TARSKI:def 2;
        then 3 <= 2 by A2,A6,A10,CARD_2:58;
        hence contradiction;
      end;
      {u, v} c= I by A3,A4,TARSKI:def 2; then
      A11: I = {u, v} by A7; then
      A12: {u, v} is linearly-independent by A1,VECTSP_7:def 3;
      Lin{u, v} = (Omega).V by A1,A11,VECTSP_7:def 3;
      hence
      ex u, v being VECTOR of V st u <> v & {u, v} is linearly-independent &
      (Omega).V = Lin{u, v} by A6,A12;
    end;
    given u, v being VECTOR of V such that
    A13: u <> v and
    A14: {u, v} is linearly-independent and
    A15: (Omega).V = Lin{u, v};
    A16: {u, v} is Basis of V by A14,A15,VECTSP_7:def 3;
    card {u, v} = 2 by A13,CARD_2:57;
    hence thesis by A16,ZMODUL03:def 5;
  end;
