reserve C for Simple_closed_curve,
  p,q,p1 for Point of TOP-REAL 2,
  i,j,k,n for Nat,
  r,s for Real;

theorem Th33:
  for S being Segmentation of C, e being Real
  st for i holds diameter Segm(S,i) < e holds diameter S < e
proof
  let S be Segmentation of C, e be Real such that
A1: for i holds diameter Segm(S,i) < e;
  consider S1 being non empty finite Subset of REAL such that
A2: S1 = { diameter Segm(S,i) where i is Element of NAT: i in dom S} and
A3: diameter S = max S1 by Def6;
  diameter S in S1 by A3,XXREAL_2:def 8;
  then ex i being Element of NAT
    st diameter S = diameter Segm(S,i) & i in dom S by A2;
  hence thesis by A1;
end;
