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 Th186:
  G is loopless implies G is G.tChromaticNum()-tcolorable
proof
  assume 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 A2: G.order()+`G.size() in TColSet(G) by A1;
  now
    let a be set;
    assume a in TColSet(G);
    then consider c being cardinal Subset of G.order()+`G.size() such that
      A3: a = c & G is c-tcolorable;
    thus a is cardinal number by A3;
  end;
  then consider c being Cardinal such that
    A4: c in TColSet(G) & c = G.tChromaticNum() by A2, GLIBPRE0:14;
  consider c9 being cardinal Subset of G.order()+`G.size() such that
    A5: c = c9 & G is c9-tcolorable by A4;
  thus thesis by A4, A5;
end;
