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

theorem Th37:
  the_Vertices_of G.allSG() = (bool the_Vertices_of G) \ {{}}
proof
  now
    let x be object;
    hereby
      assume x in the_Vertices_of G.allSG();
      then consider H being _Graph such that
        A1: H in G.allSG() & x = the_Vertices_of H by GLIB_014:def 14;
      H is Subgraph of G by A1, Th1;
      then the_Vertices_of H c= the_Vertices_of G by GLIB_000:def 32;
      then A2: x in bool the_Vertices_of G by A1;
      not x in {{}} by A1, TARSKI:def 1;
      hence x in (bool the_Vertices_of G) \ {{}} by A2, XBOOLE_0:def 5;
    end;
    reconsider X = x as set by TARSKI:1;
    assume x in (bool the_Vertices_of G) \ {{}};
    then x in bool the_Vertices_of G & not x in {{}} by XBOOLE_0:def 5;
    then reconsider X as non empty Subset of the_Vertices_of G by TARSKI:def 1;
    set S = the Function of {},X;
    set H = createGraph(X,{},S,S);
    the_Vertices_of H = X & H in G.allSG() by Lm1;
    hence x in the_Vertices_of G.allSG() by GLIB_014:def 14;
  end;
  hence thesis by TARSKI:2;
end;
