
theorem Th11:
  for G being _finite _Graph, S being VNumberingSeq of G, n being
Nat st n < S.Lifespan() holds S.PickedAt(n) in dom (S.(n+1)) & dom (S.(n+1)) =
  dom (S.n) \/ {S.PickedAt(n)}
proof
  let G being _finite _Graph, GS be VNumberingSeq of G, n be Nat such that
A1: n < GS.Lifespan();
  set f = (GS.PickedAt(n)) .--> (GS.Lifespan() -' n);
  set CN1 = GS.(n+1);
  set CSN = GS.n;
  set VLN = CSN;
  set VN1 = CN1;
A2: dom f = {GS.PickedAt(n)};
A3: GS.PickedAt(n) in {GS.PickedAt(n)} by TARSKI:def 1;
A4: VN1 = VLN +* (GS.PickedAt(n) .--> (GS.Lifespan() -' n)) by A1,Def9;
  then dom VN1 = dom VLN \/ {GS.PickedAt(n)} by A2,FUNCT_4:def 1;
  hence GS.PickedAt(n) in dom CN1 by A3,XBOOLE_0:def 3;
  thus thesis by A4,A2,FUNCT_4:def 1;
end;
