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 Th9:
  g in ConwayDay(alpha) iff
    for x st x in the_Options_of g
      ex beta st beta in alpha & x in ConwayDay(beta)
proof
  hereby
    assume g in ConwayDay(alpha);
    then consider w being strict left-right such that
A1:   g = w & for x st x in (the LeftOptions of w) \/ (the RightOptions of w)
        ex beta st beta in alpha & x in ConwayDay(beta) by Th1;
    let x;
A2: the LeftOptions of w = the_LeftOptions_of g
      & the RightOptions of w = the_RightOptions_of g
      by A1,Def6,Def7;
    assume x in the_Options_of g;
    hence ex beta st beta in alpha & x in ConwayDay(beta) by A1,A2;
  end;
  hereby
    assume
A3:   for x st x in the_Options_of g
        ex beta st beta in alpha & x in ConwayDay(beta);

    ex w being strict left-right st g = w &
      for x st x in (the LeftOptions of w) \/ (the RightOptions of w)
        ex beta st beta in alpha & x in ConwayDay(beta)
    proof
      reconsider w = g as strict left-right by Th4;
      take w;
      the LeftOptions of w = the_LeftOptions_of g
        & the RightOptions of w = the_RightOptions_of g
        by Def6,Def7;
      hence thesis by A3;
    end;
    hence g in ConwayDay(alpha) by Th1;
  end;
end;
