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
  ConwayZero in the_Tree_of g
proof
  defpred Bad[ConwayGame] means not ConwayZero in the_Tree_of $1;
  assume not thesis;
  then
A1: ex g st Bad[g];
  consider g such that
A2: Bad[g] & for gO st gO in the_Options_of g holds not Bad[gO]
    from ConwayGameMin(A1);
  per cases by Th6;
    suppose g = ConwayZero;
      hence contradiction by Th24,A2;
    end;
    suppose the_Options_of g <> {};
      then consider x being object such that
A3:     x in the_Options_of g by XBOOLE_0:def 1;
      reconsider gO = x as ConwayGame by Th17,A3;
A4:   ConwayZero in the_Tree_of gO by A2,A3;
      the_Options_of g c= the_Tree_of g by Th34;
      then the_Tree_of gO c= the_Tree_of g by Th31,A3;
      hence contradiction by A2,A4;
    end;
end;
