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 being Element of dom T holds
  (T with-replacement (p,T9)).p = T9.{}
proof
  let T,T9 be DecoratedTree, p be Element of dom T;
  p in dom T with-replacement (p,dom T9) by TREES_1:def 9;
  then
A1: ex r being FinSequence of NAT st ( r in dom T9)&( p = p^r)&(
  (T with-replacement (p,T9)).p = T9.r) by TREES_2:def 11;
  p = p^{} by FINSEQ_1:34;
  hence thesis by A1,FINSEQ_1:33;
end;
