
theorem Th12:
  for F being Field for V being finite-dimensional VectSp of F for
  v being Vector of V st v <> 0.V holds dim Lin{v} = 1
proof
  let F be Field;
  let V be finite-dimensional VectSp of F;
  let v be Vector of V;
  assume v <> 0.V;
  then
A1: v <> 0.Lin{v} by VECTSP_4:11;
  v in {v} by TARSKI:def 1;
  then v in Lin{v} by VECTSP_7:8;
  then reconsider v0=v as Vector of Lin{v};
  Lin{v0}=Lin{v} & (Omega).Lin{v0} = Lin{v0} by VECTSP_9:17;
  hence thesis by A1,VECTSP_9:30;
end;
