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

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