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 Th15:
  for C being Simple_closed_curve, A being non empty Subset of
TOP-REAL 2 st A is_an_arc_of W-min C, E-max C & A c= C holds A = Lower_Arc C or
  A = Upper_Arc C
proof
  let C be Simple_closed_curve, A be non empty Subset of TOP-REAL 2 such that
A1: A is_an_arc_of W-min C, E-max C and
A2: A c= C;
  A is compact by A1,JORDAN5A:1;
  hence thesis by A1,A2,TOPMETR3:15;
end;
