theorem Th14:
  gO in the_Options_of g implies ConwayRank(gO) in ConwayRank(g)
proof
  set alpha = ConwayRank(g);
A1:
  g in ConwayDay(alpha) by Def9;
  assume gO in the_Options_of g;
  then consider beta such that
A2: beta in alpha & gO in ConwayDay(beta) by A1,Th9;
  ConwayRank(gO) c= beta by A2,Th12;
  hence thesis by A2,ORDINAL1:12;
end;
