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

theorem
  the_Vertices_of G.allInducedSG() = (bool the_Vertices_of G) \ {{}}
proof
  the_Vertices_of G.allInducedSG() c= the_Vertices_of G.allSG()
    by GLIBPRE1:115;
  then A1: the_Vertices_of G.allInducedSG() c= (bool the_Vertices_of G) \ {{}}
    by Th37;
  now
    let x be object;
    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 H = the plain inducedSubgraph of G,X;
    the_Vertices_of H = X & H in G.allInducedSG() by GLIB_000:def 37;
    hence x in the_Vertices_of G.allInducedSG() by GLIB_014:def 14;
  end;
  then (bool the_Vertices_of G)\{{}} c= the_Vertices_of G.allInducedSG()
    by TARSKI:def 3;
  hence thesis by A1, XBOOLE_0:def 10;
end;
