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;
