reserve G for _Graph;

theorem
  for G being non _trivial connected _Graph
  holds field VertexDomRel(G) = the_Vertices_of G
proof
  let G be non _trivial connected _Graph;
  now
    let C be Component of G;
    set v = the Vertex of C;
    the_Vertices_of C c= the_Vertices_of G;
    then v in the_Vertices_of G & G is Component of G
      by GLIB_002:30, TARSKI:def 3;
    then C == G by GLIB_002:34;
    hence C is non _trivial by GLIB_000:89;
  end;
  hence thesis by Th6;
end;
