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 v being object, G1 being addAdjVertexAll of G2,v,V
  holds G1.tChromaticNum() c= G2.tChromaticNum() +` 1 +` card V
proof
  let v be object, G1 be addAdjVertexAll of G2,v,V;
  per cases;
  suppose G2 is non loopless;
    then G1.tChromaticNum() = 0;
    hence thesis by XBOOLE_1:2;
  end;
  suppose G2 is loopless;
    then G1 is (G2.tChromaticNum()+`1+`card V)-tcolorable by Th186, Th175;
    hence thesis by Th189;
  end;
end;
