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
  for G1 being addVertices of G2, V
  holds G1.vChromaticNum() = G2.vChromaticNum()
proof
  let G1 be addVertices of G2, V;
  per cases;
  suppose G2 is non loopless;
    then G2.vChromaticNum() = 0 & G1.vChromaticNum() = 0;
    hence thesis;
  end;
  suppose A1: G2 is loopless;
    now
      G1 is G1.vChromaticNum()-vcolorable by A1, Th54;
      hence G2 is G1.vChromaticNum()-vcolorable by Th34;
      let c;
      assume G2 is c-vcolorable;
      then G1 is c-vcolorable by Th34;
      hence G1.vChromaticNum() c= c by Th57;
    end;
    hence thesis by Th58;
  end;
end;
