
theorem
  for G being simple complete _Graph, v being Vertex of G
  holds v.degree()+`1 = G.order()
proof
  let G be simple complete _Graph, v be Vertex of G;
  v in {v} by TARSKI:def 1;
  then not v in the_Vertices_of G \ {v} by XBOOLE_0:def 5;
  then A1: not v in v.allNeighbors() by Th45;
  thus v.degree()+`1 = v.degree() +` card {v} by CARD_1:30
    .= card v.allNeighbors() +` card {v} by GLIB_000:111
    .= card (v.allNeighbors() \/ {v}) by A1, ZFMISC_1:50, CARD_2:35
    .= card (the_Vertices_of G \ {v} \/ {v}) by Th45
    .= G.order() by ZFMISC_1:116;
end;
