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 Th163:
  G is c1-vcolorable c2-ecolorable implies G is (c1+`c2)-tcolorable
proof
  assume A1: G is c1-vcolorable c2-ecolorable;
  then consider f9 being VColoring of G such that
    A2: f9 is proper & card rng f9 c= c1;
  consider g being proper EColoring of G such that
    A3: card rng g c= c2 by A1;
  consider f being VColoring of G such that
    A4: f is proper & rng f misses rng g & card rng f9 = card rng f
    by A2, Th150;
  reconsider t = [f,g] as TColoring of G;
  A5: t is proper by A4, Th147;
  card((rng t_V)\/rng t_E) = card rng t_V +` card rng t_E by A4, CARD_2:35;
  then card((rng t_V)\/rng t_E) c= c1 +` c2
    by A2, A3, A4, CARD_2:83;
  hence thesis by A5;
end;
