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

theorem Th57:
  G is edgeless iff G.allInducedSG() = the set of all createGraph(V)
    where V is non empty Subset of the_Vertices_of G
proof
  set S = the set of all createGraph(V)
    where V is non empty Subset of the_Vertices_of G;
  hereby
    assume A1: G is edgeless;
    G.allInducedSG() c= G.allSG();
    then A2: G.allInducedSG() c= S by A1, Th9;
    now
      let x be object;
      assume x in S;
      then consider V being non empty Subset of the_Vertices_of G such that
        A3: x = createGraph(V);
      set H = createGraph(V);
      A4: G.edgesBetween(V) = the_Edges_of H by A1;
      A5: the_Vertices_of H = V;
      H is inducedSubgraph of G,V by A4, A5, GLIB_000:def 37;
      hence x in G.allInducedSG() by A3, Th45;
    end;
    then S c= G.allInducedSG() by TARSKI:def 3;
    hence G.allInducedSG() = S by A2, XBOOLE_0:def 10;
  end;
  assume A6: G.allInducedSG() = S;
  G | _GraphSelectors in G.allInducedSG() by Th47;
  then consider V being non empty Subset of the_Vertices_of G such that
    A7: G | _GraphSelectors = createGraph(V) by A6;
  A8: G == G | _GraphSelectors by GLIB_000:128;
  the_Edges_of (G | _GraphSelectors) = {} by A7;
  then the_Edges_of G = {} by A8, GLIB_000:def 34;
  hence G is edgeless;
end;
