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 Th28:
  gO in the_Tree_of g implies (gO = g or ConwayRank(gO) in ConwayRank(g))
proof
  assume gO in the_Tree_of g;
  then consider f being ConwayGameChain such that
A1: f.1 = gO & f.(len f) = g by Def12;

  reconsider n = 1 as Element of dom f by FINSEQ_5:6;
  reconsider m = len f as Element of dom f by FINSEQ_5:6;

A2:
  m >= 1 by NAT_1:14;
  per cases by A2,XXREAL_0:1;
    suppose m = 1;
      hence thesis by A1;
    end;
    suppose m > 1;
      then ConwayRank(f.n) in ConwayRank(f.m) by Th21;
      hence thesis by A1;
    end;
end;
