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
  for p1,p2,q being Point of TOP-REAL 2 st q in C & p1 in C & p2 in C &
  p1 <> p2 & p1 <> q & p2 <> q holds p1,p2 are_neighbours_wrt q,q, C
proof
  let p1,p2,q be Point of TOP-REAL 2 such that
A1: q in C and
A2: p1 in C & p2 in C & p1 <> p2 and
A3: p1 <> q & p2 <> q;
  consider P1,P2 being non empty Subset of TOP-REAL 2 such that
A4: P1 is_an_arc_of p1,p2 and
A5: P2 is_an_arc_of p1,p2 and
A6: C = P1 \/ P2 and
A7: P1 /\ P2 = {p1,p2} by A2,TOPREAL2:5;
  per cases by A1,A6,XBOOLE_0:def 3;
  suppose
A8: q in P1 & not q in P2;
    take P2;
    thus P2 is_an_arc_of p1,p2 by A5;
    thus P2 c= C by A6,XBOOLE_1:7;
    thus P2 misses {q,q} by A8,ZFMISC_1:51;
  end;
  suppose
A9: q in P2 & not q in P1;
    take P1;
    thus P1 is_an_arc_of p1,p2 by A4;
    thus P1 c= C by A6,XBOOLE_1:7;
    thus P1 misses {q,q} by A9,ZFMISC_1:51;
  end;
  suppose
    q in P1 & q in P2;
    then q in P1 /\ P2 by XBOOLE_0:def 4;
    hence thesis by A3,A7,TARSKI:def 2;
  end;
end;
