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
  for G1 being addVertices of G2, V
  holds G1.tChromaticNum() = G2.tChromaticNum()
proof
  let G1 be addVertices of G2, V;
  per cases;
  suppose G2 is non loopless;
    then G1.tChromaticNum() = 0 & G2.tChromaticNum() = 0;
    hence thesis;
  end;
  suppose G2 is loopless;
    then G1 is G2.tChromaticNum()-tcolorable &
      for c st G1 is c-tcolorable holds G2.tChromaticNum() c= c
      by Th169, Th186, Th189;
    hence thesis by Th190;
  end;
end;
