reserve c,c1,c2 for Cardinal, G,G1,G2 for _Graph, v for Vertex of G;

theorem Th26:
  for n being Nat, G1 being simple vertex-finite n-regular _Graph
  for G2 being GraphComplement of G1
  holds G2 is (G1.order()-'(n+1))-regular
proof
  let n be Nat, G1 be simple vertex-finite n-regular _Graph;
  let G2 be GraphComplement of G1;
  let v2 be Vertex of G2;
  reconsider v1 = v2 as Vertex of G1 by GLIB_012:98;
  v1.degree() < G1.order() by GLIBPRE1:114;
  then n < G1.order() by Def4;
  then A1: n+1 <= G1.order() by NAT_1:13;
  thus v2.degree() = G1.order()-(v1.degree()+1) by GLIBPRE1:112
    .= G1.order()+0-(n+1) by Def4
    .= G1.order()-'(n+1) by A1, NAT_D:37;
end;
