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 Th162:
  G is c-tcolorable implies G is c-vcolorable & G is c-ecolorable
proof
  assume G is c-tcolorable;
  then consider t being TColoring of G such that
    A1: t is proper & card((rng t_V)\/rng t_E) c= c;
  card rng t_V c= card((rng t_V)\/rng t_E) by XBOOLE_1:7, CARD_1:11;
  hence G is c-vcolorable by A1, XBOOLE_1:1;
  card rng t_E c= card((rng t_V)\/rng t_E) by XBOOLE_1:7, CARD_1:11;
  hence G is c-ecolorable by A1, XBOOLE_1:1;
end;
