
theorem Th21:
  for G being _finite _Graph, S being VNumberingSeq of G, m, n
being Nat st n < S.Lifespan() & n < m holds S.PickedAt(n) in dom (S.m) & (S.m).
  (S.PickedAt(n)) = S.Lifespan() -' n
proof
  let G be _finite _Graph, S be VNumberingSeq of G, m,n be Nat such that
A1: n < S.Lifespan() and
A2: n < m;
  set CN1 = S.(n+1);
  set VN1 = CN1;
  set v = S.PickedAt(n);
A3: VN1.v = S.Lifespan() -' n by A1,Th12;
  set CSM = S.m;
  set VLM = CSM;
  n+1 <= m by A2,NAT_1:13;
  then VN1 c= VLM by Th17;
  then
A4: dom VN1 c= dom VLM by RELAT_1:11;
  v in dom CN1 by A1,Th11;
  hence thesis by A3,A4,Th19;
end;
