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 = 1 iff ex v st v <> 0.V & (Omega).V = Lin{v}
proof
  hereby
    consider I being finite Subset of V such that
A1: I is Basis of V by Def1;
    assume dim V = 1;
    then card I = 1 by A1,Def2;
    then consider v being object such that
A2: I = {v} by CARD_2:42;
    v in I by A2,TARSKI:def 1;
    then reconsider v as VECTOR of V;
    {v} is linearly-independent by A1,A2,RLVECT_3:def 3;
    then
A3: v <> 0.V by RLVECT_3:8;
    Lin{v} = the RLSStruct of V by A1,A2,RLVECT_3:def 3;
    hence ex v st v <> 0.V & (Omega).V = Lin{v} by A3,RLSUB_1:def 4;
  end;
  given v such that
A4: v <> 0.V & (Omega).V = Lin{v};
  {v} is linearly-independent & Lin{v} = the RLSStruct of V by A4,RLSUB_1:def 4
,RLVECT_3:8;
  then
A5: {v} is Basis of V by RLVECT_3:def 3;
  card {v} = 1 by CARD_1:30;
  hence thesis by A5,Def2;
end;
