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 Th19:
  for w being strict left-right holds w is ConwayGame iff
    for z st z in (the LeftOptions of w) \/ (the RightOptions of w)
      holds z is ConwayGame
proof
  let w be strict left-right;
  hereby
    assume w is ConwayGame;
    then reconsider g = w as ConwayGame;
    the LeftOptions of w = the_LeftOptions_of g
      & the RightOptions of w = the_RightOptions_of g
      by Def6,Def7;
    then
      (the LeftOptions of w) \/ (the RightOptions of w) = the_Options_of g;
    hence for z st z in (the LeftOptions of w) \/ (the RightOptions of w)
      holds z is ConwayGame by Th17;
  end;
  hereby
    assume
A1:   for z st z in (the LeftOptions of w) \/ (the RightOptions of w)
        holds z is ConwayGame;
    set Z = the set of all  ConwayRank(z) where
        z is Element of (the LeftOptions of w) \/ (the RightOptions of w);
    set alpha = sup Z;

    now
      let z;
      assume
A2:     z in (the LeftOptions of w) \/ (the RightOptions of w);
      then ConwayRank(z) in Z;
      then ConwayRank(z) in On Z & On Z c= alpha
        by ORDINAL1:def 9,ORDINAL2:def 3;
      then
A3:     ConwayRank(z) c= alpha by ORDINAL1:def 2;

      take beta = alpha;
      thus beta in succ alpha by ORDINAL1:6;
      z is ConwayGame by A1,A2;
      hence z in ConwayDay(beta) by A3,Th12;
    end;
    then w in ConwayDay(succ alpha) by Th1;
    hence w is ConwayGame;
  end;
end;
