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 Th61:
  for T being Tree, p being FinSequence holds p in T iff <*0*>^p in ^T
proof
  let T be Tree, p be FinSequence;
  thus p in T implies <*0*>^p in ^T by Th60;
  assume <*0*>^p in ^T;
  then consider n,q such that
  n < len <*T*> and
A1: q in <*T*>.(n+1) and
A2: <*0*>^p = <*n*>^q by Def15;
A3: (<*0*>^p).1 = 0 by FINSEQ_1:41;
A4: (<*n*>^q).1 = n by FINSEQ_1:41;
  then p = q by A2,A3,FINSEQ_1:33;
  hence thesis by A1,A2,A3,A4;
end;
