
theorem Th12:
  for G being _finite _Graph, S being VNumberingSeq of G, n being
  Nat st n < S.Lifespan() holds (S.(n+1)).(S.PickedAt(n)) = S.Lifespan()-'n
proof
  let G being _finite _Graph, GS be VNumberingSeq of G, n be Nat such that
A1: n < GS.Lifespan();
  set w = GS.PickedAt(n);
  set CN1 = GS.(n+1);
  set CSN = GS.n;
  set VLN = CSN;
  set VN1 = CN1;
  set f = w .--> (GS.Lifespan() -' n);
A2: f.w = GS.Lifespan() -' n by FUNCOP_1:72;
A3: w in dom f by TARSKI:def 1;
  VN1 = VLN +* (w .--> (GS.Lifespan() -' n)) by A1,Def9;
  hence thesis by A3,A2,FUNCT_4:13;
end;
