
theorem Th36:
  for G being _finite _Graph, n being Nat holds dom ((LexBFS:CSeq(G
  )).n)`1 = the_Vertices_of(G) iff G.order() <= n
proof
  let G be _finite _Graph, n be Nat;
  set CS = LexBFS:CSeq(G);
  set CSN = CS.n;
  set VLN = CSN`1;
  set CSO = CS.G.order();
  set VLO = CSO`1;
  thus not dom VLN = the_Vertices_of G or not n < G.order() by Th32;
  card dom VLO = card the_Vertices_of G by Th32;
  then
A1: dom VLO = the_Vertices_of G by CARD_2:102;
  assume G.order() <= n;
  hence thesis by A1,Th34;
end;
