reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;

theorem
  for G1, G2 being _Graph, V being set st G1 == G2 & V c= the_Vertices_of G2
  holds G1 is addVertices of G2, V
proof
  let G1, G2 be _Graph, V be set;
  assume A1: G1 == G2 & V c= the_Vertices_of G2;
  then the_Vertices_of G1 = the_Vertices_of G2 &
    the_Edges_of G1 = the_Edges_of G2 & the_Source_of G1 = the_Source_of G2 &
    the_Target_of G1 = the_Target_of G2 by GLIB_000:def 34;
  then A3: the_Vertices_of G1 = the_Vertices_of G2 \/ V &
    the_Edges_of G1 = the_Edges_of G2 & the_Source_of G1 = the_Source_of G2 &
    the_Target_of G1 = the_Target_of G2 by A1, XBOOLE_1:12;
  G1 is Supergraph of G2 by A1, Th63;
  hence thesis by A3, Def10;
end;
