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

theorem
  G is edgeless iff G.allSpanningSG() = {G | _GraphSelectors}
proof
  hereby
    assume A1: G is edgeless;
    now
      let x be object;
      hereby
        assume x in G.allSpanningSG();
        then reconsider H = x as plain spanning Subgraph of G by Th60;
        the_Edges_of H = {} by A1;
        then the_Edges_of H = the_Edges_of G by A1;
        then G == H & G == G | _GraphSelectors by GLIB_000:128, GLIB_000:118;
        hence x = G | _GraphSelectors by GLIB_000:85, GLIB_009:44;
      end;
      assume x = G | _GraphSelectors;
      hence x in G.allSpanningSG() by Th62;
    end;
    hence G.allSpanningSG() = {G | _GraphSelectors} by TARSKI:def 1;
  end;
  assume A2: G.allSpanningSG() = {G | _GraphSelectors};
  set H = the plain removeEdges of G, the_Edges_of G;
  H in G.allSpanningSG() by Th60;
  then A3: H = G | _GraphSelectors by A2, TARSKI:def 1;
  G == G | _GraphSelectors by GLIB_000:128;
  then the_Edges_of G = the_Edges_of H by A3, GLIB_000:def 34
    .= {};
  hence thesis;
end;
