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 Th188:
  G.tChromaticNum() c= G.order()+`G.size()
proof
  per cases;
  suppose G is non loopless;
    then G.tChromaticNum() = 0;
    hence thesis by XBOOLE_1:2;
  end;
  suppose G is loopless;
    then G is G.order()-vcolorable & G is G.size()-ecolorable by Th29, Th100;
    then A1: G is (G.order()+`G.size())-tcolorable by Th163;
    G.order()+`G.size() c= G.order()+`G.size();
    then G.order()+`G.size() in TColSet(G) by A1;
    hence G.tChromaticNum() c= G.order()+`G.size() by SETFAM_1:3;
  end;
end;
