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 Th31:
  for H being Subgraph of G holds G is c-vcolorable implies H is c-vcolorable
proof
  let H be Subgraph of G;
  assume G is c-vcolorable;
  then consider f being VColoring of G such that
    A1: f is proper & card rng f c= c;
  reconsider f9 = f | the_Vertices_of H as VColoring of H by Th3;
  card rng f9 c= card rng f by RELAT_1:70, CARD_1:11;
  then card rng f9 c= c by A1, XBOOLE_1:1;
  hence thesis by A1, Th15;
end;
