
theorem
  for V being finite-dimensional RealUnitarySpace holds dim V = 1 iff ex
  v being VECTOR of V st v <> 0.V & (Omega).V = Lin{v}
proof
  let V be finite-dimensional RealUnitarySpace;
  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,RUSUB_3:def 2;
    then
A3: v <> 0.V by RLVECT_3:8;
    Lin{v} = the UNITSTR of V by A1,A2,RUSUB_3:def 2;
    hence ex v being VECTOR of V st v <> 0.V & (Omega).V = Lin{v} by A3,
RUSUB_1:def 3;
  end;
  given v being VECTOR of V such that
A4: v <> 0.V & (Omega).V = Lin{v};
  {v} is linearly-independent & Lin{v} = the UNITSTR of V by A4,RLVECT_3:8
,RUSUB_1:def 3;
  then
A5: {v} is Basis of V by RUSUB_3:def 2;
  card {v} = 1 by CARD_1:30;
  hence thesis by A5,Def2;
end;
