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

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