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;

theorem Th101:
  G is edgeless iff G is 0-ecolorable
proof
  hereby
    assume G is edgeless;
    then G.size() = 0;
    hence G is 0-ecolorable by Th100;
  end;
  assume G is 0-ecolorable;
  then consider g being proper EColoring of G such that
    A1: card rng g c= 0;
  card rng g = 0 by A1, XBOOLE_1:3;
  then g = {};
  then {} = dom g
    .= the_Edges_of G by PARTFUN1:def 2;
  hence thesis;
end;
