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

theorem
  G is connected iff G | _GraphSelectors in G.allConnectedSG()
proof
  hereby
    assume A1: G is connected;
    G is Subgraph of G by GLIB_000:40;
    hence G | _GraphSelectors in G.allConnectedSG() by A1, Th125;
  end;
  assume G | _GraphSelectors in G.allConnectedSG();
  then A2: G | _GraphSelectors is connected by Th124;
  G | _GraphSelectors == G by GLIB_000:128;
  hence thesis by A2, GLIB_002:8;
end;
