reserve C, P for Simple_closed_curve,
  a, b, c, d, e for Point of TOP-REAL 2;

theorem
  LE a,W-min(P),P implies a in Lower_Arc(P)
proof
  assume
A1: LE a,W-min(P),P;
  per cases by A1,JORDAN6:def 10;
  suppose
    a in Upper_Arc(P) & W-min(P) in Lower_Arc(P) & not W-min(P) =
W-min(P) or a in Lower_Arc(P) & W-min(P) in Lower_Arc(P) & not W-min(P) = W-min
    (P) & LE a,W-min(P),Lower_Arc(P),E-max(P),W-min(P);
    hence thesis;
  end;
  suppose that
A2: a in Upper_Arc(P) and
    W-min(P) in Upper_Arc(P) and
A3: LE a,W-min(P),Upper_Arc(P),W-min(P),E-max(P);
    Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by JORDAN6:def 8;
    then consider
    f being Function of I[01], (TOP-REAL 2)|Upper_Arc(P), r being
    Real such that
A4: f is being_homeomorphism and
A5: f.0 = W-min(P) and
A6: f.1 = E-max(P) and
A7: 0 <= r and
A8: r <= 1 and
A9: f.r = a by A2,Th1;
    thus thesis
    proof
      per cases;
      suppose
        r = 0;
        hence thesis by A5,A9,JORDAN7:1;
      end;
      suppose
        r <> 0;
        then r > 0 by A7,XXREAL_0:1;
        hence thesis by A3,A4,A5,A6,A8,A9,JORDAN5C:def 3;
      end;
    end;
  end;
end;
