reserve x,y,z for object, X,Y for set,
  i,k,n for Nat,
  p,q,r,s for FinSequence,
  w for FinSequence of NAT,
  f for Function;

theorem Th53:
  p is Tree-yielding implies elementary_tree len p c= tree(p)
proof
  assume
A1: p is Tree-yielding;
  then
A2: rng p is constituted-Trees;
  let q be object;
  assume q in elementary_tree len p;
  then q in {<*n*>: n < len p} or q in {{}} by XBOOLE_0:def 3;
  then
A3: (ex n st q = <*n*> & n < len p) or q = {} by TARSKI:def 1;
  assume
A4: not thesis;
  then consider n such that
A5: q = <*n*> and
A6: n < len p by A3,TREES_1:22;
  p.(n+1) in rng p by A6,Lm3;
  then reconsider t = p.(n+1) as Tree by A2;
A7: {} in t by TREES_1:22;
  <*n*>^{} = q by A5,FINSEQ_1:34;
  hence thesis by A1,A4,A6,A7,Def15;
end;
