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 Th13:
  ConwayRank(g) in alpha iff ex beta st beta in alpha & g in ConwayDay(beta)
proof
  hereby
    assume
A1:   ConwayRank(g) in alpha;
    take beta = ConwayRank(g);
    thus beta in alpha by A1;
    thus g in ConwayDay(beta) by Th12;
  end;
 thus (ex beta st beta in alpha & g in ConwayDay(beta))
    implies ConwayRank(g) in alpha by Th12,ORDINAL1:12;
end;
