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
  S-bound C <= (LMP Lower_Arc C)`2
proof
  set w = (E-bound C + W-bound C) / 2;
A1: Lower_Arc C /\ Vertical_Line w c= C /\ Vertical_Line w by JORDAN6:61
,XBOOLE_1:26;
  proj2.:(Lower_Arc C /\ Vertical_Line w) is non empty & proj2.:(C /\
  Vertical_Line w) is bounded_below by Th13,Th22;
  then
A2: lower_bound(proj2.:(Lower_Arc C /\ Vertical_Line w)) >=
 lower_bound(proj2.:(C /\
  Vertical_Line w)) by A1,RELAT_1:123,SEQ_4:47;
  W-bound C = W-bound Lower_Arc C & E-bound C = E-bound Lower_Arc C by Th19
,Th20;
  then
A3: (LMP Lower_Arc C)`2 = lower_bound(proj2.:(Lower_Arc C /\ Vertical_Line w))
 by EUCLID:52;
  (LMP C)`2 = lower_bound(proj2.:(C /\ Vertical_Line w)) & S-bound C
   <= (LMP C)`2
  by Th40,EUCLID:52;
  hence thesis by A2,A3,XXREAL_0:2;
end;
