
theorem Th39:
  for G being _finite _Graph holds (LexBFS:CSeq(G))``1.Lifespan() =
  (LexBFS:CSeq(G)).Lifespan()
proof
  let G be _finite _Graph;
  set S = LexBFS:CSeq(G);
  set VN = S``1;
  set ls = G.order();
A1: VN is eventually-constant by Th38;
A2: (S.(ls+1))`1 = S``1.(ls+1) by Def15;
A3: now
    let n be Nat such that
A4: VN.n = VN.(n+1) and
A5: ls > n;
    n+1 <= ls by A5,NAT_1:13;
    then
A6: card dom (S.(n+1))`1 = n+1 by Th32;
A7: (S.(n+1))`1 = VN.(n+1) by Def15;
A8: (S.n)`1 = VN.n by Def15;
    card dom (S.n)`1 = n by A5,Th32;
    hence contradiction by A4,A6,A8,A7;
  end;
  (S.ls)`1 = S``1.ls by Def15;
  then
A9: VN.ls = VN.(ls+1) by A2,Th33,NAT_1:11;
  S.Lifespan() = ls by Th37;
  hence thesis by A1,A9,A3,GLIB_000:def 56;
end;
