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 Th22:
  for f being ConwayGameChain
    for n,m being Element of dom f st n <= m holds
      ConwayRank(f.n) c= ConwayRank(f.m)
proof
  let f be ConwayGameChain;
  let n,m be Element of dom f such that
A1: n <= m;
  per cases by A1,XXREAL_0:1;
    suppose n < m;
      then ConwayRank(f.n) in ConwayRank(f.m) by Th21;
      hence thesis by ORDINAL1:def 2;
    end;
    suppose n = m;
      hence thesis;
    end;
end;
