reserve x,y,y1,y2,z,e,s for set;
reserve alpha,beta,gamma for Ordinal;
reserve n,m,k for Nat;
reserve g,g0,g1,g2,gO,gL,gR,gLL,gLR,gRL,gRR for ConwayGame;

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;
