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 Th12:
  g in ConwayDay(alpha) iff ConwayRank(g) c= alpha
proof
  hereby
    assume
A1:   g in ConwayDay(alpha);
    assume not ConwayRank(g) c= alpha;
    then alpha in ConwayRank(g) by ORDINAL1:16;
    hence contradiction by A1,Def9;
  end;
  hereby
    assume ConwayRank(g) c= alpha;
    then
A2:   ConwayDay(ConwayRank(g)) c= ConwayDay(alpha) by Th3;
    g in ConwayDay(ConwayRank(g)) by Def9;
    hence g in ConwayDay(alpha) by A2;
  end;
end;
