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 G2 being non edgeless _Graph, v,e,w being object
  for G1 being addAdjVertex of G2,v,e,w
  holds G1.tChromaticNum() c= G2.tChromaticNum() +` 1
proof
  let G2 be non edgeless _Graph, v,e,w be object;
  let G1 be addAdjVertex of G2,v,e,w;
  per cases;
  suppose G1 is non loopless;
    then G1.tChromaticNum() = 0;
    hence thesis by XBOOLE_1:2;
  end;
  suppose G1 is loopless;
    then G2 is loopless;
    then A1: G2 is G2.tChromaticNum()-tcolorable by Th186;
    G1 is (G2.tChromaticNum()+`1)-tcolorable by A1, Th172;
    hence thesis by Th189;
  end;
end;
