
theorem
  for G being vertex-finite simple _Graph
  holds G.size() <= (G.order()^2 - G.order())/2
proof
  let G be vertex-finite simple _Graph;
  consider f being one-to-one Function such that
    A1: dom f = the_Edges_of G & rng f c= 2Set the_Vertices_of G and
    for e being object st e in dom f holds
      f.e = {(the_Source_of G).e,(the_Target_of G).e} by Th5;
  reconsider n = G.order() - 1 as Nat by CHORD:1;
  A2: card 2Set the_Vertices_of G
     = (card the_Vertices_of G) choose 2 by GLIBPRE0:20
    .= n * (n+1) / 2 by NUMPOLY1:72
    .= (G.order()*G.order() - G.order())/2
    .= (G.order()^2 - G.order())/2 by SQUARE_1:def 1;
  card rng f c= card 2Set the_Vertices_of G by A1, CARD_1:11;
  then Segm G.size() c= Segm card 2Set the_Vertices_of G by A1, CARD_1:70;
  hence thesis by A2, NAT_1:39;
end;
