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

theorem
  G.allInducedSG() /\ G.allSpanningSG() = {G | _GraphSelectors}
proof
  now
    let x be object;
    hereby
      assume x in G.allInducedSG() /\ G.allSpanningSG();
      then A1: x in G.allInducedSG() & x in G.allSpanningSG()
        by XBOOLE_0:def 4;
      then consider V being non empty Subset of the_Vertices_of G such that
        A2: x is plain inducedSubgraph of G, V by Th45;
      reconsider H = x as plain inducedSubgraph of G, V by A2;
      A3: H is spanning by A1, Th60;
      then A4: the_Vertices_of G = the_Vertices_of H by GLIB_000:def 33
        .= V by GLIB_000:def 37;
      the_Edges_of G = G.edgesBetween(the_Vertices_of G) by GLIB_000:34
        .= the_Edges_of H by A4, GLIB_000:def 37;
      then G == H & G == G | _GraphSelectors by A3, GLIB_000:128, GLIB_000:118;
      hence x = G | _GraphSelectors by GLIB_000:85, GLIB_009:44;
    end;
    assume x = G | _GraphSelectors;
    then x in G.allInducedSG() & x in G.allSpanningSG()
      by Th47, Th62;
    hence x in G.allInducedSG() /\ G.allSpanningSG()
      by XBOOLE_0:def 4;
  end;
  hence thesis by TARSKI:def 1;
end;
