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 Th29:
  gO in the_Tree_of g implies ConwayRank(gO) c= ConwayRank(g)
proof
  assume
A1: gO in the_Tree_of g;
  per cases by A1,Th28;
    suppose gO = g;
      hence thesis;
    end;
    suppose
      ConwayRank(gO) in ConwayRank(g);
      hence thesis by ORDINAL1:def 2;
    end;
end;
