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

theorem
  G is edgeless iff G.allSG() = G.allInducedSG()
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;
    hence G.allSG() = S by Th9
      .= G.allInducedSG() by A1, Th57;
  end;
  assume A2: G.allSG() = G.allInducedSG();
  set H = createGraph([#]the_Vertices_of G);
  consider V being non empty Subset of the_Vertices_of G such that
    A3: H is plain inducedSubgraph of G,V by A2, Lm1, Th45;
  A4: V = the_Vertices_of H by A3, GLIB_000:def 37
    .= the_Vertices_of G;
  {} = the_Edges_of H
    .= G.edgesBetween(V) by A3, GLIB_000:def 37
    .= the_Edges_of G by A4, GLIB_000:34;
  hence thesis;
end;
