
theorem Th45:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is weak_SG-embedding
  holds G1.order() c= G2.order() & G1.size() c= G2.size()
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume F is weak_SG-embedding;
  then A1: dom F_V = the_Vertices_of G1 & dom F_E = the_Edges_of G1 &
    F_V is one-to-one & F_E is one-to-one by Def11;
  rng F_V c= the_Vertices_of G2 & rng F_E c= the_Edges_of G2;
  then A2: card the_Vertices_of G1 c= card the_Vertices_of G2 &
    card the_Edges_of G1 c= card the_Edges_of G2 by A1, CARD_1:10;
  then G1.order() c= card the_Vertices_of G2 by GLIB_000:def 24;
  hence G1.order() c= G2.order() by GLIB_000:def 24;
  G1.size() c= card the_Edges_of G2 by A2, GLIB_000:def 25;
  hence G1.size() c= G2.size() by GLIB_000:def 25;
end;
