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

theorem Th49:
  for n being Nat, G1 being Dsimple vertex-finite n-Dregular _Graph
  for G2 being DGraphComplement of G1
  holds G2 is (G1.order()-'(n+1))-Dregular
proof
  let n be Nat, G1 be Dsimple vertex-finite n-Dregular _Graph;
  let G2 be DGraphComplement of G1;
  let v2 be Vertex of G2;
  reconsider v1 = v2 as Vertex of G1 by GLIB_012:80;
  v1.inDegree() < G1.order() by GLIBPRE1:113;
  then n < G1.order() by Def8;
  then A1: n+1 <= G1.order() by NAT_1:13;
  thus v2.inDegree() = G1.order()-(v1.inDegree()+1) by GLIBPRE1:111
    .= G1.order()+0-(n+1) by Def8
    .= G1.order()-'(n+1) by A1, NAT_D:37;
  thus v2.outDegree() = G1.order()-(v1.outDegree()+1) by GLIBPRE1:111
    .= G1.order()+0-(n+1) by Def8
    .= G1.order()-'(n+1) by A1, NAT_D:37;
end;
