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 G2 being removeParallelEdges of G1
  holds G1.vChromaticNum() = G2.vChromaticNum()
proof
  let G2 be removeParallelEdges of G1;
  per cases;
  suppose G1 is non loopless;
    then G1.vChromaticNum() = 0 & G2.vChromaticNum() = 0;
    hence thesis;
  end;
  suppose G1 is loopless;
    then G2 is G1.vChromaticNum()-vcolorable & for c st G2 is c-vcolorable
      holds G1.vChromaticNum() c= c by Th40, Th54, Th57;
    hence thesis by Th58;
  end;
end;
