reserve
  X,x,y,z for set,
  k,n,m for Nat ,
  f for Function,
  p,q,r for FinSequence of NAT;
reserve p1,p2,p3 for FinSequence;
reserve T,T1 for Tree;
reserve fT,fT1 for finite Tree;
reserve t for Element of T;
reserve w for FinSequence;
reserve t1,t2 for Element of T;

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;
