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 Th42:
  for F being PGraphMapping of G1, G2
  st F is weak_SG-embedding & G2 is c-vcolorable holds G1 is c-vcolorable
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is weak_SG-embedding & G2 is c-vcolorable;
  then consider f2 being VColoring of G2 such that
    A2: f2 is proper & card rng f2 c= c;
  reconsider f1 = f2*F_V as VColoring of G1 by A1, Th9;
  card rng f1 c= card rng f2 by RELAT_1:26, CARD_1:11;
  then card rng f1 c= c by A2, XBOOLE_1:1;
  hence thesis by A1, A2, Th26;
end;
