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 Th75:
  for F being PGraphMapping of G1, G2
  st F is isomorphism holds G1.vChromaticNum() = G2.vChromaticNum()
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is isomorphism;
  per cases;
  suppose G1 is non loopless;
    then G1 is non loopless & G2 is non loopless by A1, GLIB_010:35;
    then G1.vChromaticNum() = 0 & G2.vChromaticNum() = 0;
    hence thesis;
  end;
  suppose A2: G1 is loopless;
    now
      G1 is G1.vChromaticNum()-vcolorable by A2, Th54;
      hence G2 is G1.vChromaticNum()-vcolorable by A1, Th43;
      let c;
      assume G2 is c-vcolorable;
      then G1 is c-vcolorable by A1, Th43;
      hence G1.vChromaticNum() c= c by Th57;
    end;
    hence thesis by Th58;
  end;
end;
