
theorem
  for G being vertex-finite simple _Graph, v being Vertex of G
  holds v.degree() < G.order()
proof
  let G be vertex-finite simple _Graph, v be Vertex of G;
  A1: G.order()-1 is Nat by CHORD:1;
  not v in v.allNeighbors() by GLIB_000:112;
  then card v.allNeighbors() <= card(the_Vertices_of G \ {v})
    by NAT_1:43, ZFMISC_1:34;
  then v.degree() <= card(the_Vertices_of G \ {v}) by GLIB_000:111;
  then v.degree() <= G.order() - card {v} by CARD_2:44;
  then v.degree() <= G.order() - 1 by CARD_1:30;
  then v.degree() < G.order() - 1 + 1 by A1, NAT_1:13;
  hence v.degree() < G.order();
end;
