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 Th23:
  for f being ConwayGameChain st f.(len f) in ConwayDay(alpha)
    holds f.1 in ConwayDay(alpha)
proof
  let f be ConwayGameChain;
  assume
A1: f.(len f) in ConwayDay(alpha);

  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;

  n <= m by NAT_1:14;
  then ConwayRank(f.n) c= ConwayRank(f.m)
    & ConwayRank(f.m) c= alpha by Th22,A1,Th12;
  then ConwayRank(f.n) c= alpha;
  hence f.1 in ConwayDay(alpha) by Th12;
end;
