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 Th54:
  G is loopless implies G is G.vChromaticNum()-vcolorable
proof
  assume G is loopless;
  then G is G.order()-vcolorable & G.order() c= G.order() by Th29;
  then A1: G.order() in VColSet(G);
  now
    let a be set;
    assume a in VColSet(G);
    then consider c being cardinal Subset of G.order() such that
      A2: a = c & G is c-vcolorable;
    thus a is cardinal number by A2;
  end;
  then consider c being Cardinal such that
    A3: c in VColSet(G) & c = G.vChromaticNum() by A1, GLIBPRE0:14;
  consider c9 being cardinal Subset of G.order() such that
    A4: c = c9 & G is c9-vcolorable by A3;
  thus thesis by A3, A4;
end;
