reserve x, y, z for object,
  i, j, n for Nat,
  p, q, r for FinSequence,
  v for FinSequence of NAT;
reserve T,T9 for DecoratedTree,
  x,y for set;

theorem
  not x in rng T or not x in Leaves T implies (T,x) <- T9 = T
proof
A1: Leaves T c= rng T by RELAT_1:111;
  assume not x in rng T or not x in Leaves T;
then A2: not x in Leaves T by A1;
  thus
A3: dom ((T,x) <- T9) = dom T
  proof
    let p be FinSequence of NAT;
 p in dom (T,x) <- T9 iff p in dom T or
    ex q being Node of T, r being Node of T9 st
    q in Leaves dom T & T.q = x & p = q^r by Def7;
    hence thesis by A2,FUNCT_1:def 6;
  end;
  let p be Node of (T,x) <- T9;
  reconsider p9 = p as Node of T by A3;
 p9 in Leaves dom T implies T.p9 in Leaves T by FUNCT_1:def 6;
  hence thesis by A2,Def7;
end;
