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

theorem
  a <> c & a <> d & b <> d & a,b,c,d are_in_this_order_on P & a,c,b,d
  are_in_this_order_on P implies b = c
proof
  assume that
A1: a <> c and
A2: a <> d and
A3: b <> d and
A4: a,b,c,d are_in_this_order_on P and
A5: a,c,b,d are_in_this_order_on P;
  per cases by A4;
  suppose
A6: LE a,b,P & LE b,c,P & LE c,d,P;
    then
A7: LE b,d,P by JORDAN6:58;
    thus thesis
    proof
      per cases by A5;
      suppose
        LE a,c,P & LE c,b,P & LE b,d,P;
        hence thesis by A6,JORDAN6:57;
      end;
      suppose
        LE c,b,P & LE b,d,P & LE d,a,P;
        hence thesis by A6,JORDAN6:57;
      end;
      suppose
A8:     LE b,d,P & LE d,a,P & LE a,c,P;
        LE a,d,P by A6,A7,JORDAN6:58;
        hence thesis by A2,A8,JORDAN6:57;
      end;
      suppose
        LE d,a,P & LE a,c,P & LE c,b,P;
        hence thesis by A6,JORDAN6:57;
      end;
    end;
  end;
  suppose
A9: LE b,c,P & LE c,d,P & LE d,a,P;
    then
A10: LE c,a,P by JORDAN6:58;
    thus thesis
    proof
      per cases by A5;
      suppose
        LE a,c,P & LE c,b,P & LE b,d,P;
        hence thesis by A9,JORDAN6:57;
      end;
      suppose
        LE c,b,P & LE b,d,P & LE d,a,P;
        hence thesis by A9,JORDAN6:57;
      end;
      suppose
        LE b,d,P & LE d,a,P & LE a,c,P;
        hence thesis by A1,A10,JORDAN6:57;
      end;
      suppose
        LE d,a,P & LE a,c,P & LE c,b,P;
        hence thesis by A9,JORDAN6:57;
      end;
    end;
  end;
  suppose
A11: LE c,d,P & LE d,a,P & LE a,b,P;
    then
A12: LE c,a,P by JORDAN6:58;
    thus thesis
    proof
      per cases by A5;
      suppose
        LE a,c,P & LE c,b,P & LE b,d,P;
        hence thesis by A1,A12,JORDAN6:57;
      end;
      suppose
A13:    LE c,b,P & LE b,d,P & LE d,a,P;
        LE d,b,P by A11,JORDAN6:58;
        hence thesis by A3,A13,JORDAN6:57;
      end;
      suppose
        LE b,d,P & LE d,a,P & LE a,c,P;
        hence thesis by A1,A12,JORDAN6:57;
      end;
      suppose
        LE d,a,P & LE a,c,P & LE c,b,P;
        hence thesis by A1,A12,JORDAN6:57;
      end;
    end;
  end;
  suppose
A14: LE d,a,P & LE a,b,P & LE b,c,P;
    thus thesis
    proof
      per cases by A5;
      suppose
        LE a,c,P & LE c,b,P & LE b,d,P;
        hence thesis by A14,JORDAN6:57;
      end;
      suppose
        LE c,b,P & LE b,d,P & LE d,a,P;
        hence thesis by A14,JORDAN6:57;
      end;
      suppose
A15:    LE b,d,P & LE d,a,P & LE a,c,P;
        LE d,b,P by A14,JORDAN6:58;
        hence thesis by A3,A15,JORDAN6:57;
      end;
      suppose
        LE d,a,P & LE a,c,P & LE c,b,P;
        hence thesis by A14,JORDAN6:57;
      end;
    end;
  end;
end;
