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 Th6:
  g = ConwayZero iff the_Options_of g = {}
proof
  hereby
    assume g = ConwayZero;
    then the_LeftOptions_of g = {} & the_RightOptions_of g = {};
    hence the_Options_of g = {};
  end;
  hereby
    reconsider w = g as strict left-right by Th4;
    assume the_Options_of g = {};
    then the_LeftOptions_of g = {} & the_RightOptions_of g = {};
    then the LeftOptions of w = {} & the RightOptions of w = {}
      by Def6,Def7;
    hence g = ConwayZero;
  end;
end;
