
theorem
  for G1 being vertex-finite simple _Graph, G2 being GraphComplement of G1
  for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2
  holds v2.degree() = G1.order() - (v1.degree()+1)
proof
  let G1 be vertex-finite simple _Graph, G2 be GraphComplement of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  v1.allNeighbors() c= the_Vertices_of G1;
  then v1.allNeighbors() c= the_Vertices_of G2 by GLIB_012:98;
  then A2: v1.allNeighbors() \/ {v2} c= the_Vertices_of G2 by XBOOLE_1:8;
  A3: not v2 in v1.allNeighbors() by A1, GLIB_000:112;
  thus v2.degree() = card v2.allNeighbors() by GLIB_000:111
    .= card (the_Vertices_of G2 \(v1.allNeighbors()\/{v2})) by A1, GLIB_012:118
    .= G2.order() - card (v1.allNeighbors()\/{v2}) by A2, CARD_2:44
    .= G2.order() - (card v1.allNeighbors() + 1) by A3, CARD_2:41
    .= G1.order() - (card v1.allNeighbors() + 1) by GLIB_012:98
    .= G1.order() - (v1.degree()+1) by GLIB_000:111;
end;
