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 Th169:
  for G1 being addVertices of G2, V
  holds G1 is c-tcolorable iff G2 is c-tcolorable
proof
  let G1 be addVertices of G2, V;
  G2 is Subgraph of G1 by GLIB_006:57;
  hence G1 is c-tcolorable implies G2 is c-tcolorable by Th166;
  assume G2 is c-tcolorable;
  then consider t2 being TColoring of G2 such that
    A1: t2 is proper & card((rng t2_V)\/rng t2_E) c= c;
  set x = the Element of rng t2_V;
  set h = (V \ the_Vertices_of G2) --> x, f = t2_V +* h;
  A2: dom h = V \ the_Vertices_of G2;
  then reconsider t1 = [f, t2_E] as TColoring of G1 by Th139;
  t1_E = t2_E & t1_V = t2_V +* h;
  then A3: t1 is proper by A1, A2, Th154;
  (rng f)\/rng t1_E c= (rng t2_V)\/rng t2_E by Lm3, XBOOLE_1:9;
  then card((rng f)\/rng t1_E) c= card((rng t2_V)\/rng t2_E) by CARD_1:11;
  hence thesis by A1, A3, XBOOLE_1:1;
end;
