reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;

theorem
  card ([#]V) = 2 implies dim V = 1
proof
  assume
A1: card ([#]V) = 2;
  then
A2: [#]V is finite;
  reconsider C = [#]V as finite set by A1;
  reconsider V as finite-dimensional VectSp of F by A2,Th4;
  ex v being Vector of V st v <> 0.V & (Omega).V = Lin ({v})
  proof
    consider x,y being object such that
A3: x <> y and
A4: [#]V = {x,y} by A1,CARD_2:60;
    per cases by A4,TARSKI:def 2;
    suppose
A5:   x = 0.V;
      then reconsider x as Element of V;
      reconsider y as Element of V by A4,TARSKI:def 2;
      set L = Lin ({y});
      take y;
      for v being Element of V holds v in (Omega).V iff v in L
      proof
        let v be Element of V;
        v in (Omega).V implies v in L
        proof
          assume v in (Omega).V;
A6:       y in {y} by TARSKI:def 1;
A7:       0.L in L;
          per cases by A4,TARSKI:def 2;
          suppose
            v = x;
            hence thesis by A5,A7,VECTSP_4:def 2;
          end;
          suppose
            v = y;
            hence thesis by A6,VECTSP_7:8;
          end;
        end;
        hence thesis;
      end;
      hence thesis by A3,A5,VECTSP_4:30;
    end;
    suppose
A8:   y = 0.V;
      reconsider x as Element of V by A4,TARSKI:def 2;
      reconsider y as Element of V by A8;
      set L = Lin ({x});
      take x;
      for v being Element of V holds v in (Omega).V iff v in L
      proof
        let v be Element of V;
        v in (Omega).V implies v in L
        proof
          assume v in (Omega).V;
A9:      x in {x} by TARSKI:def 1;
A10:      0.L in L;
          per cases by A4,TARSKI:def 2;
          suppose
            v = y;
            hence thesis by A8,A10,VECTSP_4:def 2;
          end;
          suppose
            v = x;
            hence thesis by A9,VECTSP_7:8;
          end;
        end;
        hence thesis;
      end;
      hence thesis by A3,A8,VECTSP_4:30;
    end;
  end;
  hence thesis by VECTSP_9:30;
end;
