
theorem Th69:
  for G being _finite _Graph, n being Nat holds dom ((MCS:CSeq(G)).
  n)`1 = the_Vertices_of G iff G.order() <= n
proof
  let G be _finite _Graph, n be Nat;
  set VLN = ((MCS:CSeq(G)).n)`1;
  set CSO = (MCS:CSeq(G)).G.order();
  set VLO = CSO`1;
  thus dom VLN = the_Vertices_of G implies not n < G.order() by Th65;
  card dom VLO = card the_Vertices_of G by Th65;
  then
A1: dom VLO = the_Vertices_of G by CARD_2:102;
  assume G.order() <= n;
  hence thesis by A1,Th67;
end;
