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 Th166:
  for H being Subgraph of G holds G is c-tcolorable implies H is c-tcolorable
proof
  let H be Subgraph of G;
  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;
  reconsider t9 = [ t_V | the_Vertices_of H , t_E | the_Edges_of H ]
    as TColoring of H by Th138;
  rng t9_V c= rng t_V & rng t9_E c= rng t_E by RELAT_1:70;
  then (rng t9_V)\/rng t9_E c= (rng t_V)\/rng t_E by XBOOLE_1:13;
  then card((rng t9_V)\/rng t9_E) c= card((rng t_V)\/rng t_E) by CARD_1:11;
  then card((rng t9_V)\/rng t9_E) c= c by A1, XBOOLE_1:1;
  hence thesis by A1, Th151;
end;
