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

theorem Th110:
  for F being PGraphMapping of G1, G2
  st F is weak_SG-embedding & G2 is c-ecolorable holds G1 is c-ecolorable
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is weak_SG-embedding & G2 is c-ecolorable;
  then consider g2 being proper EColoring of G2 such that
    A2: card rng g2 c= c;
  reconsider g1 = g2*(F_E) as EColoring of G1 by A1, Th84;
  A3: g1 is proper by A1, Th98;
  card rng g1 c= card rng g2 by RELAT_1:26, CARD_1:11;
  hence thesis by A2, A3, XBOOLE_1:1;
end;
