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 Th187:
  G is non loopless iff G.tChromaticNum() = 0
proof
  hereby
    assume A1: G is non loopless;
    now
      given x being object such that
        A2: x in TColSet(G);
      consider c being cardinal Subset of G.order()+`G.size() such that
        A3: x = c & G is c-tcolorable by A2;
      thus contradiction by A1, A3;
    end;
    hence G.tChromaticNum() = 0 by XBOOLE_0:def 1, SETFAM_1:1;
  end;
  assume A4: G.tChromaticNum() = 0 & G is loopless;
  per cases;
  suppose A5: TColSet(G) = {};
    G is G.order()-vcolorable & G is G.size()-ecolorable by A4, Th29, Th100;
    then A6: 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 A6;
    hence contradiction by A5;
  end;
  suppose A7: TColSet(G) <> {};
    now
      let a be set;
      assume a in TColSet(G);
      then consider c being cardinal Subset of G.order()+`G.size() such that
        A8: a = c & G is c-tcolorable;
      thus a is cardinal number by A8;
    end;
    then consider c being Cardinal such that
      A9: c in TColSet(G) & c = G.tChromaticNum() by A7, GLIBPRE0:14;
    consider c9 being cardinal Subset of G.order()+`G.size() such that
      A10: c = c9 & G is c9-tcolorable by A9;
    thus contradiction by A4, A9, A10;
  end;
end;
