
theorem Th9:
  for Gs being iterative eventually-constant ManySortedSet of NAT
  for n being Nat st Gs.Lifespan() <= n holds Gs.(Gs.Lifespan()) = Gs.n
proof
  let Gs be iterative eventually-constant ManySortedSet of NAT;
  set GL = Gs.Lifespan();
  defpred P[Nat] means Gs.GL = Gs.(GL+$1);
  let n be Nat;
  assume GL <= n;
  then
A1: ex i being Nat st GL + i = n by NAT_1:10;
A2: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume P[k];
    then Gs.(GL+1) = Gs.(GL+k+1) by Def5;
    hence thesis by GLIB_000:def 56;
  end;
A3: P[ 0 ];
  for k being Nat holds P[k] from NAT_1:sch 2(A3,A2);
  hence thesis by A1;
end;
