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

theorem Th22:
  for G being edgeless _Graph, S being GraphUnionSet, G9 being GraphUnion of S
  st (for v being Vertex of G ex H9 being Element of S st
      v in the_Vertices_of H9)
  holds G is Subgraph of G9
proof
  let G be edgeless _Graph, S be GraphUnionSet, G9 be GraphUnion of S;
  assume A1: for v being Vertex of G ex H9 being Element of S st
    v in the_Vertices_of H9;
  now
    let x be object;
    assume x in the_Vertices_of G;
    then consider H9 being Element of S such that
      A2: x in the_Vertices_of H9 by A1;
    H9 is Subgraph of G9 by GLIB_014:21;
    then the_Vertices_of H9 c= the_Vertices_of G9 by GLIB_000:def 32;
    hence x in the_Vertices_of G9 by A2;
  end;
  hence thesis by TARSKI:def 3, GLIBPRE1:68;
end;
