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 Th57:
  G is c-vcolorable implies G.vChromaticNum() c= c
proof
  assume A1: G is c-vcolorable;
  per cases;
  suppose not G.order() in c;
    then c c= G.order() by ORDINAL1:16;
    then c in VColSet(G) by A1;
    hence thesis by SETFAM_1:3;
  end;
  suppose G.order() in c;
    then G.vChromaticNum() in c by Th56, ORDINAL1:12;
    hence thesis by ORDINAL1:def 2;
  end;
end;
