theorem Th23:
  for f holds {LSeg(f,i): 1<=i & i+1<=len f}
     is finite
proof
  let f;
  set F = {LSeg(f,i): 1<=i & i+1<=len f},
F9 = {LSeg(f,i): 1<=i & i<=len f};
  F c= F9
  proof
    let x be object;
    assume x in F;
    then consider i being Nat such that
A1: x = LSeg(f,i) & 1<=i and
A2: i+1<=len f;
    i <= i + 1 by NAT_1:11;
    then i <= len f by A2,XXREAL_0:2;
    hence thesis by A1;
  end;
  hence thesis by Th22,FINSET_1:1;
end;
