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 Th55:
  G is non loopless iff G.vChromaticNum() = 0
proof
  hereby
    assume A1: G is non loopless;
    now
      given x being object such that
        A2: x in VColSet(G);
      consider c being cardinal Subset of G.order() such that
        A3: x = c & G is c-vcolorable by A2;
      thus contradiction by A1, A3;
    end;
    hence G.vChromaticNum() = 0 by XBOOLE_0:def 1, SETFAM_1:1;
  end;
  assume A4: G.vChromaticNum() = 0 & G is loopless;
  per cases;
  suppose A5: VColSet(G) = {};
    G is G.order()-vcolorable & G.order() c= G.order() by A4, Th29;
    then G.order() in VColSet(G);
    hence contradiction by A5;
  end;
  suppose A6: VColSet(G) <> {};
    now
      let a be set;
      assume a in VColSet(G);
      then consider c being cardinal Subset of G.order() such that
        A7: a = c & G is c-vcolorable;
      thus a is cardinal number by A7;
    end;
    then consider c being Cardinal such that
      A8: c in VColSet(G) & c = G.vChromaticNum() by A6, GLIBPRE0:14;
    consider c9 being cardinal Subset of G.order() such that
      A9: c = c9 & G is c9-vcolorable by A8;
    thus contradiction by A4, A8, A9;
  end;
end;
