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

theorem
  G | _GraphSelectors is_maximal_in SubgraphRel(G)
proof
  now
    G | _GraphSelectors in G.allSG() by Th3;
    hence A1: G | _GraphSelectors in field SubgraphRel(G) by Th40;
    given y being set such that
      A2: y in field SubgraphRel(G) & y <> G | _GraphSelectors and
      A3: [G | _GraphSelectors, y] in SubgraphRel(G);
    y in G.allSG() by A2, Th40;
    then reconsider H = y as plain Subgraph of G by Th1;
    H = H | _GraphSelectors by GLIB_000:128, GLIB_009:44;
    then [H, G | _GraphSelectors] in SubgraphRel(G) by Th39;
    hence contradiction by A1, A2, A3, RELAT_2:def 4, RELAT_2:def 12;
  end;
  hence thesis by ORDERS_1:def 12;
end;
