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;
reserve t for TColoring of G;

theorem Th176:
  for F being PGraphMapping of G1, G2
  st F is weak_SG-embedding & G2 is c-tcolorable holds G1 is c-tcolorable
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is weak_SG-embedding & G2 is c-tcolorable;
  then consider t2 being TColoring of G2 such that
    A2: t2 is proper & card((rng t2_V)\/rng t2_E) c= c;
  reconsider t1 = [ t2_V*(F_V) , t2_E*(F_E) ] as TColoring of G1 by A1, Th145;
  A3: t1 is proper by A1, A2, Th160;
  rng t1_V c= rng t2_V & rng t1_E c= rng t2_E by RELAT_1:26;
  then (rng t1_V)\/rng t1_E c= (rng t2_V)\/rng t2_E by XBOOLE_1:13;
  then card((rng t1_V)\/rng t1_E) c= card((rng t2_V)\/rng t2_E) by CARD_1:11;
  hence thesis by A2, A3, XBOOLE_1:1;
end;
