
theorem Th13:
  for G being Dsimple Dcomplete _Graph, v being Vertex of G
  holds v.inDegree()+`1 = G.order() & v.outDegree()+`1 = G.order()
proof
  let G be Dsimple Dcomplete _Graph, v be Vertex of G;
  v in {v} by TARSKI:def 1;
  then A1: not v in the_Vertices_of G \ {v} by XBOOLE_0:def 5;
  then not v in v.inNeighbors() by Th12;
  then A2: v.inNeighbors() misses {v} by ZFMISC_1:50;
  thus v.inDegree()+`1 = v.inDegree() +` card {v} by CARD_1:30
    .= card v.inNeighbors() +` card {v} by GLIB_000:109
    .= card (v.inNeighbors() \/ {v}) by A2, CARD_2:35
    .= card (the_Vertices_of G \ {v} \/ {v}) by Th12
    .= G.order() by ZFMISC_1:116;
  not v in v.outNeighbors() by A1, Th12;
  then A3: v.outNeighbors() misses {v} by ZFMISC_1:50;
  thus v.outDegree()+`1 = v.outDegree() +` card {v} by CARD_1:30
    .= card v.outNeighbors() +` card {v} by GLIB_000:110
    .= card (v.outNeighbors() \/ {v}) by A3, CARD_2:35
    .= card (the_Vertices_of G \ {v} \/ {v}) by Th12
    .= G.order() by ZFMISC_1:116;
end;
