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

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