reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring of G;

theorem Th52:
  for F being PGraphMapping of G1,G2
  st F is weak_SG-embedding & G2 is finite-vcolorable
  holds G1 is finite-vcolorable
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is weak_SG-embedding & G2 is finite-vcolorable;
  then consider n such that
    A2: G2 is n-vcolorable;
  G1 is n-vcolorable by A1, A2, Th42;
  hence thesis;
end;
