reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem Th12:
  for C being compact Subset of TOP-REAL 2 holds
  north_halfline UMP C \ {UMP C} c= UBD C
proof
  let C be compact Subset of TOP-REAL 2;
  set A = north_halfline UMP C \ {UMP C};
  reconsider A as non bounded Subset of T2 by JORDAN2C:122,TOPREAL6:90;
  A is convex by JORDAN21:6;
  hence thesis by Th10,JORDAN2C:125;
end;
