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

theorem Th99:
  for G2, V for G1 being addVertices of G2, V
  holds G1.size() = G2.size() &
    G1.order() = G2.order() +` card (V \ the_Vertices_of G2)
proof
  let G2, V;
  let G1 be addVertices of G2, V;
  thus G1.size() = card the_Edges_of G1 by GLIB_000:def 25
    .= card the_Edges_of G2 by Def10
    .= G2.size() by GLIB_000:def 25;
  thus G1.order() = card the_Vertices_of G1 by GLIB_000:def 24
    .= card (the_Vertices_of G2 \/ V) by Def10
    .= card (the_Vertices_of G2 \/ (V \ the_Vertices_of G2)) by XBOOLE_1:39
    .= card the_Vertices_of G2 +` card (V \ the_Vertices_of G2)
      by XBOOLE_1:79, CARD_2:35
    .= G2.order() +` card (V \ the_Vertices_of G2) by GLIB_000:def 24;
end;
