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 Th190:
  (G is c-tcolorable & for d being Cardinal st G is d-tcolorable holds c c= d)
  implies G.tChromaticNum() = c
proof
  assume G is c-tcolorable;
  then A1: G.tChromaticNum() c= c & G is loopless by Th189;
  assume A2: for d being Cardinal st G is d-tcolorable holds c c= d;
  G is G.order()-vcolorable & G is G.size()-ecolorable by A1, Th29, Th100;
  then A3: G is (G.order()+`G.size())-tcolorable by Th163;
  G.order()+`G.size() c= G.order()+`G.size();
  then A4: G.order()+`G.size() in TColSet(G) by A3;
  now
    let x be set;
    assume x in TColSet(G);
    then consider d being cardinal Subset of G.order()+`G.size() such that
      A5: x = d & G is d-tcolorable;
    thus c c= x by A2, A5;
  end;
  then c c= G.tChromaticNum() by A4, SETFAM_1:5;
  hence thesis by A1, XBOOLE_0:def 10;
end;
