reserve n for Element of NAT,
  V for Subset of TOP-REAL n,
  s,s1,s2,t,t1,t2 for Point of TOP-REAL n,
  C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  a,p ,p1,p2,q,q1,q2 for Point of TOP-REAL 2;

theorem
  p1 in C & p2 in C & q1 in C & p1 <> p2 & q1 <> p1 & q1 <> p2 & q2 <>
  p1 & q2 <> p2 implies p1,p2 are_neighbours_wrt q1,q2, C or p1,q1
  are_neighbours_wrt p2,q2, C
proof
  assume that
A1: p1 in C & p2 in C and
A2: q1 in C and
A3: p1 <> p2 and
A4: q1 <> p1 and
A5: q1 <> p2 and
A6: q2 <> p1 & q2 <> p2;
  consider P,P1 being non empty Subset of TOP-REAL 2 such that
A7: P is_an_arc_of p1,p2 and
A8: P1 is_an_arc_of p1,p2 and
A9: C = P \/ P1 and
A10: P /\ P1 = {p1,p2} by A1,A3,TOPREAL2:5;
A11: P c= C by A9,XBOOLE_1:7;
  assume
A12: for A being Subset of TOP-REAL 2 st A is_an_arc_of p1,p2 & A c= C
  holds A meets {q1,q2};
  then
A13: P meets {q1,q2} by A7,A9,XBOOLE_1:7;
A14: P1 c= C by A9,XBOOLE_1:7;
  per cases by A13,ZFMISC_1:51;
  suppose that
A15: q1 in P and
A16: not q2 in P;
    now
      take A = Segment(P,p1,p2,p1,q1);
A17:  now
A18:    A = {q where q is Point of TOP-REAL 2: LE p1,q,P,p1,p2 & LE q,q1,
        P,p1,p2} by JORDAN6:26;
        assume p2 in A;
        then
        ex q being Point of TOP-REAL 2 st p2 = q & LE p1,q,P,p1,p2 & LE q
        ,q1,P,p1,p2 by A18;
        hence contradiction by A5,A7,JORDAN6:55;
      end;
      LE p1, q1, P, p1, p2 by A7,A15,JORDAN5C:10;
      hence A is_an_arc_of p1,q1 by A4,A7,JORDAN16:21;
A19:  A c= P by JORDAN16:2;
      hence A c= C by A11;
      not q2 in A by A16,A19;
      hence A misses {p2,q2} by A17,ZFMISC_1:51;
    end;
    hence thesis;
  end;
  suppose that
A20: q2 in P and
A21: not q1 in P;
    now
      take A = Segment(P1,p1,p2,p1,q1);
A22:  now
A23:    A = {q where q is Point of TOP-REAL 2: LE p1,q,P1,p1,p2 & LE q,q1
        ,P1,p1,p2} by JORDAN6:26;
        assume p2 in A;
        then ex q being Point of TOP-REAL 2 st p2 = q & LE p1,q,P1,p1,p2 & LE
        q,q1,P1,p1,p2 by A23;
        hence contradiction by A5,A8,JORDAN6:55;
      end;
      q1 in P1 by A2,A9,A21,XBOOLE_0:def 3;
      then LE p1, q1, P1, p1, p2 by A8,JORDAN5C:10;
      hence A is_an_arc_of p1,q1 by A4,A8,JORDAN16:21;
A24:  A c= P1 by JORDAN16:2;
      hence A c= C by A14;
      now
        assume q2 in A;
        then q2 in {p1,p2} by A10,A20,A24,XBOOLE_0:def 4;
        hence contradiction by A6,TARSKI:def 2;
      end;
      hence A misses {p2,q2} by A22,ZFMISC_1:51;
    end;
    hence thesis;
  end;
  suppose that
A25: q1 in P & q2 in P;
    P1 meets {q1,q2} by A12,A8,A9,XBOOLE_1:7;
    then q1 in P1 or q2 in P1 by ZFMISC_1:51;
    then q1 in {p1,p2} or q2 in {p1,p2} by A10,A25,XBOOLE_0:def 4;
    hence thesis by A4,A5,A6,TARSKI:def 2;
  end;
end;
