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
  for S being Segmentation of C, i holds diameter Segm(S,i) <= diameter S
proof
  let S be Segmentation of C, i;
  consider S1 be non empty finite Subset of REAL such that
A1: S1 = { diameter Segm(S,j) where j is Element of NAT: j in dom S} and
A2: diameter S = max S1 by Def6;
  per cases;
  suppose 1 <= i & i < len S;
    then i in dom S by FINSEQ_3:25;
    then diameter Segm(S,i) in S1 by A1;
    hence thesis by A2,XXREAL_2:def 8;
  end;
  suppose
A3: not(1 <= i & i < len S);
A4: Segm(S,len S) = Segment(S/.len S,S/.1,C) by Def4
      .= Segm(S,i) by A3,Def4;
    len S in dom S by FINSEQ_5:6;
    then diameter Segm(S,i) in S1 by A1,A4;
    hence thesis by A2,XXREAL_2:def 8;
  end;
end;
