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,T9 being DecoratedTree, p,q being Element of dom T st
  not p is_a_prefix_of q holds (T with-replacement (p,T9)).q = T.q
proof
  let T,T9 be DecoratedTree, p,q be Element of dom T;
  assume
A1: not p is_a_prefix_of q;
  then
A2: q in dom T with-replacement(p,dom T9) by TREES_2:7;
  not ex r being FinSequence of NAT st r in dom T9 & q = p^r & (T
  with-replacement(p,T9)).q = T9.r by A1,TREES_1:1;
  hence thesis by A2,TREES_2:def 11;
end;
