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 Th38:
  S-bound C < (UMP C)`2
proof
  set u = UMP C, l = LMP C;
A1: now
    assume
A2: S-bound C = u`2;
    l`2 < u`2 & l in C by Th31,Th36;
    hence contradiction by A2,PSCOMP_1:24;
  end;
  u in C by Th30;
  then S-bound C <= u`2 by PSCOMP_1:24;
  hence thesis by A1,XXREAL_0:1;
end;
