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

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