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 Th25:
  for f being ConwayGameChain for n being non zero Nat holds
    f|n is ConwayGameChain
proof
  let f be ConwayGameChain;
  let n be non zero Nat;
  set ls = len (f|n);

A1:
  f|n is ConwayGame-valued
  proof
    let x such that
A2:   x in dom (f|n);
    dom (f|n) c= dom f by RELAT_1:60;
    then f.x is ConwayGame by A2;
    hence (f|n).x is ConwayGame by A2,FUNCT_1:47;
  end;
  reconsider fs = (f|n) as
    ConwayGame-valued non empty FinSequence by A1;

  fs is ConwayGameChain-like
  proof
    let n be Element of dom fs such that
A3:   n > 1;
    dom fs c= dom f & n in dom fs & n-1 in dom fs by Th20,A3,RELAT_1:60;
    then n in dom f & f.n = fs.n & f.(n-1) = fs.(n-1) by FUNCT_1:47;
    hence thesis by Def11,A3;
  end;
  hence thesis;
end;
