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 p1,p2 being Point of TOP-REAL 2 for P being Subset of TOP-REAL 2
st P c= C & P is_an_arc_of p1,p2 & W-min C in P & E-max C in P holds Upper_Arc
  C c= P or Lower_Arc C c= P
proof
  let p1,p2 be Point of TOP-REAL 2;
  let P being Subset of TOP-REAL 2 such that
A1: P c= C and
A2: P is_an_arc_of p1,p2 and
A3: W-min C in P and
A4: E-max C in P;
  reconsider P9 = P as non empty Subset of TOP-REAL 2 by A3;
A5: W-min C <> E-max C by TOPREAL5:19;
  per cases by A2,A3,A4,A5,JORDAN5C:14;
  suppose
A6: LE W-min C, E-max C, P,p1,p2;
    set S = Segment(P9,p1,p2,W-min C, E-max C);
    reconsider S9 = S as non empty Subset of TOP-REAL 2 by A6,Th5;
    S c= P by Th2;
    then S c= C by A1;
    then S9 = Lower_Arc C or S9 = Upper_Arc C by A2,A5,A6,Th15,Th21;
    hence thesis by Th2;
  end;
  suppose
A7: LE E-max C, W-min C, P,p1,p2;
    set S = Segment(P9,p1,p2,E-max C, W-min C);
    reconsider S9 = S as non empty Subset of TOP-REAL 2 by A7,Th5;
    S is_an_arc_of E-max C, W-min C by A2,A5,A7,Th21;
    then
A8: S9 is_an_arc_of W-min C, E-max C by JORDAN5B:14;
    S c= P by Th2;
    hence thesis by A1,A8,Th15,XBOOLE_1:1;
  end;
end;
