theorem Th37:
  {} is AntiChain_of_Prefixes of T & { {} } is AntiChain_of_Prefixes of T
proof
 {} is AntiChain_of_Prefixes-like;
  then reconsider S = {} as AntiChain_of_Prefixes;
 S c= T;
  hence {} is AntiChain_of_Prefixes of T by Def11;
  reconsider S = D as AntiChain_of_Prefixes by Th35;
 S is AntiChain_of_Prefixes of T
  proof
    let x be object;
    assume x in S;
then  x = {} by TARSKI:def 1;
    hence thesis by Th21;
  end;
  hence thesis;
end;
