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

theorem Th38:
  the_Edges_of G.allSG() = bool the_Edges_of G
proof
  now
    let x be object;
    hereby
      assume x in the_Edges_of G.allSG();
      then consider H being _Graph such that
        A1: H in G.allSG() & x = the_Edges_of H by GLIB_014:def 15;
      H is Subgraph of G by A1, Th1;
      then the_Edges_of H c= the_Edges_of G by GLIB_000:def 32;
      hence x in bool the_Edges_of G by A1;
    end;
    reconsider X = x as set by TARSKI:1;
    set H = the plain inducedSubgraph of G, the_Vertices_of G, X;
    assume x in bool the_Edges_of G;
    then X c= the_Edges_of G;
    then A2: X c= G.edgesBetween(the_Vertices_of G) by GLIB_000:34;
    the_Vertices_of G c= the_Vertices_of G;
    then A3: the_Edges_of H = X by A2, GLIB_000:def 37;
    H in G.allSG() by Th1;
    hence x in the_Edges_of G.allSG() by A3, GLIB_014:def 15;
  end;
  hence thesis by TARSKI:2;
end;
