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
  for T being Tree holds ^T = elementary_tree 1 with-replacement(<*0*>, T)
proof
  let T be Tree;
  let p be FinSequence of NAT;
A1: <*T*>.1 = T;
A2: len <*T*> = 1 by FINSEQ_1:40;
A3: 0+1 = 1;
A4: {} in T by TREES_1:22;
A5: <*0*> in elementary_tree 1 by TARSKI:def 2,TREES_1:51;
  thus p in ^T implies p in elementary_tree 1 with-replacement(<*0*>, T)
  proof
    assume
A6: p in ^T;
    now
      assume p <> {};
      then consider n,q such that
A7:   n < len <*T*> and
A8:   q in <*T*>.(n+1) and
A9:   p = <*n*>^q by A6,Def15;
      reconsider q as FinSequence of NAT by A9,FINSEQ_1:36;
A10:  n = 0 by A2,A3,A7,NAT_1:13;
      p = <*n*>^q by A9;
      hence thesis by A5,A8,A10,TREES_1:def 9;
    end;
    hence thesis by TREES_1:22;
  end;
  assume p in elementary_tree 1 with-replacement(<*0*>,T);
  then
A11: p in elementary_tree 1 & not <*0*> is_a_proper_prefix_of p or
  ex r being FinSequence of NAT st r in T & p = <*0*>^r by A5,TREES_1:def 9;
  now
    assume p in elementary_tree 1;
    then p = {} or p = <*0*> & p^{} = p by FINSEQ_1:34,TARSKI:def 2,TREES_1:51;
    hence thesis by A1,A2,A3,A4,Def15;
  end;
  hence thesis by A1,A2,A3,A11,Def15;
end;
