reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem Th80:
  G is acyclic iff G | _GraphSelectors in G.allForests()
proof
  hereby
    assume A1: G is acyclic;
    G is Subgraph of G by GLIB_000:40;
    hence G | _GraphSelectors in G.allForests() by A1, Th79;
  end;
  assume G | _GraphSelectors in G.allForests();
  then A2: G | _GraphSelectors is acyclic by Th78;
  G | _GraphSelectors == G by GLIB_000:128;
  hence thesis by A2, GLIB_002:44;
end;
