
theorem Th51:
  for G1, G2 being _finite _Graph, F being PGraphMapping of G1, G2
  st F is strong_SG-embedding & G1.order() = G2.order() & G1.size() = G2.size()
  holds F is isomorphism
proof
  let G1, G2 be _finite _Graph, F be PGraphMapping of G1, G2;
  assume that
    A1: F is strong_SG-embedding and
    A2: G1.order() = G2.order() & G1.size() = G2.size();
  A3: card rng F_V = card dom F_V by A1, CARD_1:70
    .= card the_Vertices_of G1 by A1, Def11
    .= G2.order() by A2, GLIB_000:def 24
    .= card the_Vertices_of G2 by GLIB_000:def 24;
  card rng F_E = card dom F_E by A1, CARD_1:70
    .= card the_Edges_of G1 by A1, Def11
    .= G2.size() by A2, GLIB_000:def 25
    .= card the_Edges_of G2 by GLIB_000:def 25;
  then F is onto by A3, CARD_2:102;
  hence thesis by A1;
end;
