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

theorem
  c <> d & a,b,c,d are_in_this_order_on P implies ex e st e <> c & e <>
  d & a,c,e,d are_in_this_order_on P
proof
  assume that
A1: c <> d and
A2: LE a,b,P & LE b,c,P & LE c,d,P or LE b,c,P & LE c,d,P & LE d,a,P or
  LE c,d,P & LE d,a,P & LE a,b,P or LE d,a,P & LE a,b,P & LE b,c,P;
  per cases by A2;
  suppose that
A3: LE a,b,P & LE b,c,P and
A4: LE c,d,P;
    consider e such that
A5: e <> c & e <> d & LE c,e,P & LE e,d,P by A1,A4,Th8;
    take e;
    LE a,c,P by A3,JORDAN6:58;
    hence thesis by A5;
  end;
  suppose that
A6: LE b,c,P & LE c,d,P and
A7: LE d,a,P;
    consider e such that
A8: e <> c & e <> d & LE c,e,P & LE e,d,P by A1,A6,Th8;
    take e;
    thus thesis by A7,A8;
  end;
  suppose that
A9: LE c,d,P and
A10: LE d,a,P and
    LE a,b,P;
    consider e such that
A11: e <> c & e <> d & LE c,e,P & LE e,d,P by A1,A9,Th8;
    take e;
    thus thesis by A10,A11;
  end;
  suppose that
A12: LE d,a,P and
A13: LE a,b,P and
A14: LE b,c,P;
    thus thesis
    proof
A15:  LE a,c,P by A13,A14,JORDAN6:58;
      per cases;
      suppose
A16:    d = W-min(P);
        c in P by A14,JORDAN7:5;
        then consider e such that
A17:    e <> c and
A18:    LE c,e,P by Th7;
        take e;
        thus e <> c by A17;
        thus e <> d by A1,A16,A18,JORDAN7:2;
        thus thesis by A12,A15,A18;
      end;
      suppose
A19:    d <> W-min(P);
        take e = W-min(P);
        d in P by A12,JORDAN7:5;
        then
A20:    LE e,d,P by JORDAN7:3;
        now
          LE d,b,P by A12,A13,JORDAN6:58;
          then
A21:      LE d,c,P by A14,JORDAN6:58;
          assume e = c;
          hence contradiction by A1,A20,A21,JORDAN6:57;
        end;
        hence e <> c;
        thus e <> d by A19;
        thus thesis by A12,A15,A20;
      end;
    end;
  end;
end;
