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 Th41:
  (LMP C)`2 < N-bound C
proof
  set u = UMP C, l = LMP C;
A1: now
    assume
A2: N-bound C = l`2;
    l`2 < u`2 & u in C by Th30,Th36;
    hence contradiction by A2,PSCOMP_1:24;
  end;
  l in C by Th31;
  then l`2 <= N-bound C by PSCOMP_1:24;
  hence thesis by A1,XXREAL_0:1;
end;
