
theorem Th31:
  for G being _finite _Graph, n being Nat st card dom ((LexBFS:CSeq
(G)).n)`1 < G.order() holds ((LexBFS:CSeq(G)).(n+1))`1 = ((LexBFS:CSeq(G)).n)`1
  +* (LexBFS:PickUnnumbered((LexBFS:CSeq(G)).n) .--> (G.order()-'(card (dom ((
  LexBFS:CSeq(G)).n)`1))))
proof
  let G be _finite _Graph, n be Nat;
  set CS = LexBFS:CSeq(G);
  assume
A1: card dom ((CS.n)`1) < G.order();
  set CN1 = CS.(n+1);
  set CSN = CS.n;
  set VLN = CSN`1;
  set w = LexBFS:PickUnnumbered(CSN);
  CN1 = LexBFS:Step(CSN) by Def16;
  then CN1 = LexBFS:Update(CSN, w, card (dom VLN)) by A1,Def13;
  hence thesis;
end;
