
theorem Th64:
  for G being _finite _Graph, n being Nat st card dom (((MCS:CSeq(G
  )).n))`1 < G.order() holds ((MCS:CSeq(G)).(n+1))`1 = ((MCS:CSeq(G)).n)`1 +* (
MCS:PickUnnumbered((MCS:CSeq(G)).n) .--> (G.order()-'(card (dom ((MCS:CSeq(G)).
  n)`1))))
proof
  let G be _finite _Graph, n be Nat such that
A1: card (dom ((MCS:CSeq(G)).n)`1) < G.order();
  set CN1 = (MCS:CSeq(G)).(n+1);
  set CSN = (MCS:CSeq(G)).n;
  set VLN = CSN`1;
  set w = MCS:PickUnnumbered(CSN);
  set k = G.order() -' card (dom VLN);
  CN1 = MCS:Step(CSN) by Def25;
  then CN1 = MCS:Update(CSN, w, card (dom VLN)) by A1,Def22;
  then consider L being MCS:Labeling of G such that
A2: L = [CSN`1 +* (w .--> k), CSN`2] and
A3: CN1 = MCS:LabelAdjacent(L, w) by Def21;
  CN1`1 = [CSN`1 +* (w .--> k), CSN`2]`1 by A3,A2;
  hence thesis;
end;
