reserve G, G2 for _Graph, V, E for set,
  v for object;

theorem
  for G2, E for G1 being reverseEdgeDirections of G2, E
  holds G1.order() = G2.order() & G1.size() = G2.size()
proof
  let G2, E;
  let G1 be reverseEdgeDirections of G2, E;
  A1: the_Vertices_of G1 = the_Vertices_of G2 &
    the_Edges_of G1 = the_Edges_of G2 by Th4;
  thus G1.order() = card the_Vertices_of G1 by GLIB_000:def 24
    .= G2.order() by A1, GLIB_000:def 24;
  thus G1.size() = card the_Edges_of G1 by GLIB_000:def 25
    .= G2.size() by A1, GLIB_000:def 25;
end;
