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 Th56:
  G.vChromaticNum() c= G.order()
proof
  per cases;
  suppose G is non loopless;
    then G.vChromaticNum() = 0;
    hence thesis by XBOOLE_1:2;
  end;
  suppose G is loopless;
    then G is G.order()-vcolorable & G.order() c= G.order() by Th29;
    then G.order() in VColSet(G);
    hence G.vChromaticNum() c= G.order() by SETFAM_1:3;
  end;
end;
