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

theorem
  a in C & b in C & c in C & d in C implies a,b,c,d are_in_this_order_on
C or a,b,d,c are_in_this_order_on C or a,c,b,d are_in_this_order_on C or a,c,d,
  b are_in_this_order_on C or a,d,b,c are_in_this_order_on C or a,d,c,b
  are_in_this_order_on C
proof
  assume that
A1: a in C and
A2: b in C and
A3: c in C and
A4: d in C;
  per cases;
  suppose
    LE a,b,C & LE b,c,C & LE c,d,C;
    hence thesis;
  end;
  suppose
A5: LE a,b,C & LE b,c,C & not LE c,d,C;
    then
A6: LE d,c,C by A3,A4,JORDAN16:7;
    thus thesis
    proof
      per cases by A1,A4,JORDAN16:7;
      suppose
A7:     LE a,d,C;
        LE b,d,C or LE d,b,C by A2,A4,JORDAN16:7;
        hence thesis by A5,A6,A7;
      end;
      suppose
        LE d,a,C;
        hence thesis by A5;
      end;
    end;
  end;
  suppose
A8: LE a,b,C & not LE b,c,C & LE c,d,C;
    then
A9: LE c,b,C by A2,A3,JORDAN16:7;
    thus thesis
    proof
      per cases by A2,A4,JORDAN16:7;
      suppose
A10:    LE b,d,C;
        LE a,c,C or LE c,a,C by A1,A3,JORDAN16:7;
        hence thesis by A8,A9,A10;
      end;
      suppose
A11:    LE d,b,C;
        thus thesis
        proof
          per cases by A1,A3,JORDAN16:7;
          suppose
            LE a,c,C;
            hence thesis by A8,A11;
          end;
          suppose
A12:        LE c,a,C;
            LE a,d,C or LE d,a,C by A1,A4,JORDAN16:7;
            hence thesis by A8,A11,A12;
          end;
        end;
      end;
    end;
  end;
  suppose
A13: LE a,b,C & not LE b,c,C & not LE c,d,C;
    then
A14: LE d,c,C by A3,A4,JORDAN16:7;
A15: LE c,b,C by A2,A3,A13,JORDAN16:7;
    thus thesis
    proof
      per cases by A1,A3,JORDAN16:7;
      suppose
A16:    LE a,c,C;
        LE a,d,C or LE d,a,C by A1,A4,JORDAN16:7;
        hence thesis by A15,A14,A16;
      end;
      suppose
        LE c,a,C;
        hence thesis by A13,A14;
      end;
    end;
  end;
  suppose
A17: not LE a,b,C & LE b,c,C & LE c,d,C;
    then
A18: LE b,a,C by A1,A2,JORDAN16:7;
    thus thesis
    proof
      per cases by A1,A4,JORDAN16:7;
      suppose
A19:    LE a,d,C;
        LE a,c,C or LE c,a,C by A1,A3,JORDAN16:7;
        hence thesis by A17,A18,A19;
      end;
      suppose
        LE d,a,C;
        hence thesis by A17;
      end;
    end;
  end;
  suppose
A20: not LE a,b,C & LE b,c,C & not LE c,d,C;
    then
A21: LE b,a,C & LE d,c,C by A1,A2,A3,A4,JORDAN16:7;
    thus thesis
    proof
      per cases by A1,A4,JORDAN16:7;
      suppose
        LE a,d,C;
        hence thesis by A21;
      end;
      suppose
A22:    LE d,a,C;
A23:    LE b,d,C or LE d,b,C by A2,A4,JORDAN16:7;
        LE a,c,C or LE c,a,C by A1,A3,JORDAN16:7;
        hence thesis by A20,A21,A22,A23;
      end;
    end;
  end;
  suppose
A24: not LE a,b,C & not LE b,c,C & LE c,d,C;
    then
A25: LE b,a,C & LE c,b,C by A1,A2,A3,JORDAN16:7;
    thus thesis
    proof
      per cases by A1,A4,JORDAN16:7;
      suppose
        LE a,d,C;
        hence thesis by A25;
      end;
      suppose
A26:    LE d,a,C;
        LE b,d,C or LE d,b,C by A2,A4,JORDAN16:7;
        hence thesis by A24,A25,A26;
      end;
    end;
  end;
  suppose
A27: not LE a,b,C & not LE b,c,C & not LE c,d,C;
    then
A28: LE d,c,C by A3,A4,JORDAN16:7;
    LE b,a,C & LE c,b,C by A1,A2,A3,A27,JORDAN16:7;
    hence thesis by A28;
  end;
end;
