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;
reserve V for finite-dimensional RealLinearSpace,
  W, W1, W2 for Subspace of V,
  u, v for VECTOR of V;

theorem
  dim V = 2 iff ex u, v st u <> v & {u, v} is linearly-independent &
  (Omega).V = Lin{u, v}
proof
  hereby
    consider I being finite Subset of V such that
A1: I is Basis of V by Def1;
    assume dim V = 2;
    then
A2: card I = 2 by A1,Def2;
    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};
      for x be object st x in {u, v, w} holds x in I
         by A3,A4,A8,ENUMSET1:def 1;
      then {u, v, w} c= I;
      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;
    for x be object st x in {u, v} holds x in I by A3,A4,TARSKI:def 2;
    then {u, v} c= I;
    then
A11: I = {u, v} by A7,XBOOLE_0:def 10;
    then
A12: {u, v} is linearly-independent by A1,RLVECT_3:def 3;
    Lin{u, v} = the RLSStruct of V by A1,A11,RLVECT_3:def 3
      .= (Omega).V by RLSUB_1:def 4;
    hence
    ex u, v st u <> v & {u, v} is linearly-independent & (Omega).V = Lin{
    u, v} by A6,A12;
  end;
  given u, v such that
A13: u <> v and
A14: {u, v} is linearly-independent and
A15: (Omega).V = Lin{u, v};
  Lin{u, v} = the RLSStruct of V by A15,RLSUB_1:def 4;
  then
A16: {u, v} is Basis of V by A14,RLVECT_3:def 3;
  card {u, v} = 2 by A13,CARD_2:57;
  hence thesis by A16,Def2;
end;
