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