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
  G1 == G2 implies G1.vChromaticNum() = G2.vChromaticNum()
proof
  assume A1: G1 == G2;
  now
    let x be object;
    hereby
      assume x in VColSet(G1);
      then consider c being cardinal Subset of G1.order() such that
        A2: x = c & G1 is c-vcolorable;
      G1.order() = G2.order() & G2 is c-vcolorable
        by A1, A2, Th32, GLIB_000:90;
      hence x in VColSet(G2) by A2;
    end;
    assume x in VColSet(G2);
    then consider c being cardinal Subset of G2.order() such that
      A3: x = c & G2 is c-vcolorable;
    G1.order() = G2.order() & G1 is c-vcolorable by A1, A3, Th32, GLIB_000:90;
    hence x in VColSet(G1) by A3;
  end;
  hence thesis by TARSKI:2;
end;
