reserve C for Simple_closed_curve,
  A,A1,A2 for Subset of TOP-REAL 2,
  p,p1,p2,q ,q1,q2 for Point of TOP-REAL 2,
  n for Element of NAT;

theorem
  for P being Subset of TOP-REAL 2, p1,p2,q1,q2 being Point of TOP-REAL
2 st P is_an_arc_of p1,p2 & q1 in P & q2 in P & q1 <> p1 & q1 <> p2 & q2 <> p1
  & q2 <> p2 & q1 <> q2 ex Q being non empty Subset of TOP-REAL 2 st Q
  is_an_arc_of q1,q2 & Q c= P & Q misses {p1,p2}
proof
  let P be Subset of TOP-REAL 2, p1,p2,q1,q2 be Point of TOP-REAL 2 such that
A1: P is_an_arc_of p1,p2 and
A2: q1 in P & q2 in P and
A3: q1 <> p1 and
A4: q1 <> p2 and
A5: q2 <> p1 and
A6: q2 <> p2 and
A7: q1 <> q2;
  per cases by A1,A2,A7,JORDAN5C:14;
  suppose
A8: LE q1,q2,P,p1,p2;
    set S = Segment(P,p1,p2,q1,q2);
    S is_an_arc_of q1,q2 by A1,A7,A8,Th21;
    then reconsider S as non empty Subset of TOP-REAL 2 by TOPREAL1:1;
    take S;
    thus S is_an_arc_of q1,q2 by A1,A7,A8,Th21;
    thus S c= P by Th2;
    now
A9:   S = {q where q is Point of TOP-REAL 2 : LE q1,q,P,p1,p2 & LE q,q2,P
      ,p1,p2} by JORDAN6:26;
      assume
A10:  S meets {p1,p2};
      per cases by A10,ZFMISC_1:51;
      suppose
        p1 in S;
        then
        ex q being Point of TOP-REAL 2 st q = p1 & LE q1,q,P,p1,p2 & LE q
        ,q2,P,p1,p2 by A9;
        hence contradiction by A1,A3,JORDAN6:54;
      end;
      suppose
        p2 in S;
        then
        ex q being Point of TOP-REAL 2 st q = p2 & LE q1,q,P,p1,p2 & LE q
        ,q2,P,p1,p2 by A9;
        hence contradiction by A1,A6,JORDAN6:55;
      end;
    end;
    hence thesis;
  end;
  suppose
A11: LE q2,q1,P,p1,p2;
    set S = Segment(P,p1,p2,q2,q1);
    S is_an_arc_of q2,q1 by A1,A7,A11,Th21;
    then reconsider S as non empty Subset of TOP-REAL 2 by TOPREAL1:1;
    take S;
    S is_an_arc_of q2,q1 by A1,A7,A11,Th21;
    hence S is_an_arc_of q1,q2 by JORDAN5B:14;
    thus S c= P by Th2;
    now
A12:  S = {q where q is Point of TOP-REAL 2 : LE q2,q,P,p1,p2 & LE q,q1,P
      ,p1,p2} by JORDAN6:26;
      assume
A13:  S meets {p1,p2};
      per cases by A13,ZFMISC_1:51;
      suppose
        p1 in S;
        then
        ex q being Point of TOP-REAL 2 st q = p1 & LE q2,q,P,p1,p2 & LE q
        ,q1,P,p1,p2 by A12;
        hence contradiction by A1,A5,JORDAN6:54;
      end;
      suppose
        p2 in S;
        then
        ex q being Point of TOP-REAL 2 st q = p2 & LE q2,q,P,p1,p2 & LE q
        ,q1,P,p1,p2 by A12;
        hence contradiction by A1,A4,JORDAN6:55;
      end;
    end;
    hence thesis;
  end;
end;
