
theorem
  for G being vertex-finite Dsimple _Graph, v being Vertex of G
  holds v.inDegree() < G.order() & v.outDegree() < G.order()
proof
  let G be vertex-finite Dsimple _Graph, v being Vertex of G;
  A1: G.order()-1 is Nat by CHORD:1;
  not v in v.allNeighbors() by GLIB_000:112;
  then A2: not v in v.inNeighbors() & not v in v.outNeighbors()
    by XBOOLE_0:def 3;
  A3: v.inNeighbors() c= the_Vertices_of G \ {v} &
    v.outNeighbors() c= the_Vertices_of G \ {v} by A2, ZFMISC_1:34;
  then card v.inNeighbors() <= card(the_Vertices_of G \ {v}) by NAT_1:43;
  then v.inDegree() <= card(the_Vertices_of G \ {v}) by GLIB_000:109;
  then v.inDegree() <= G.order() - card {v} by CARD_2:44;
  then v.inDegree() <= G.order() - 1 by CARD_1:30;
  then v.inDegree() < G.order() - 1 + 1 by A1, NAT_1:13;
  hence v.inDegree() < G.order();
  card v.outNeighbors() <= card(the_Vertices_of G \ {v}) by A3, NAT_1:43;
  then v.outDegree() <= card(the_Vertices_of G \ {v}) by GLIB_000:110;
  then v.outDegree() <= G.order() - card {v} by CARD_2:44;
  then v.outDegree() <= G.order() - 1 by CARD_1:30;
  then v.outDegree() < G.order() - 1 + 1 by A1, NAT_1:13;
  hence v.outDegree() < G.order();
end;
