reserve C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  n for Element of NAT;
reserve D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem
  (UMP Upper_Arc C)`2 <= N-bound C
proof
  set w = (E-bound C + W-bound C) / 2;
A1: Upper_Arc C /\ Vertical_Line w c= C /\ Vertical_Line w by JORDAN6:61
,XBOOLE_1:26;
  proj2.:(Upper_Arc C /\ Vertical_Line w) is non empty & proj2.:(C /\
  Vertical_Line w) is bounded_above by Th13,Th21;
  then
A2: upper_bound(proj2.:(Upper_Arc C /\ Vertical_Line w))
 <= upper_bound(proj2.:(C /\
  Vertical_Line w)) by A1,RELAT_1:123,SEQ_4:48;
  W-bound C = W-bound Upper_Arc C & E-bound C = E-bound Upper_Arc C by Th17
,Th18;
  then
A3: (UMP Upper_Arc C)`2 = upper_bound(proj2.:(Upper_Arc C /\ Vertical_Line w))
 by EUCLID:52;
  (UMP C)`2 = upper_bound(proj2.:(C /\ Vertical_Line w)) & (UMP C)`2 <=
   N-bound C
  by Th39,EUCLID:52;
  hence thesis by A2,A3,XXREAL_0:2;
end;
