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 Th58:
  (G is c-vcolorable & for d being Cardinal st G is d-vcolorable holds c c= d)
  implies G.vChromaticNum() = c
proof
  assume G is c-vcolorable;
  then A1: G.vChromaticNum() c= c & G is loopless by Th57;
  assume A2: for d being Cardinal st G is d-vcolorable holds c c= d;
  G is G.order()-vcolorable & G.order() c= G.order() by A1, Th29;
  then A3: G.order() in VColSet(G);
  now
    let x be set;
    assume x in VColSet(G);
    then consider d being cardinal Subset of G.order() such that
      A4: x = d & G is d-vcolorable;
    thus c c= x by A2, A4;
  end;
  then c c= G.vChromaticNum() by A3, SETFAM_1:5;
  hence thesis by A1, XBOOLE_0:def 10;
end;
