
theorem Th16:
  for G being _finite _Graph, S being VNumberingSeq of G, n,m being
Nat st S.Lifespan() -' n < m & m <= S.Lifespan() ex v being Vertex of G st v in
  dom (S.n) & (S.n).v = m
proof
  let G being _finite _Graph, S be VNumberingSeq of G, n,m be Nat such that
A1: S.Lifespan() -' n < m and
A2: m <= S.Lifespan();
  set CSN = S.n;
  set VLN = CSN;
  m in (Seg S.Lifespan()) \ Seg (S.Lifespan()-'n) by A1,A2,Th3;
  then m in rng VLN by Th14;
  then consider v being object such that
A3: v in dom VLN and
A4: m = VLN.v by FUNCT_1:def 3;
   reconsider v as set by TARSKI:1;
  take v;
  thus v is Vertex of G by A3;
  thus v in dom CSN by A3;
  thus thesis by A4;
end;
