theorem Lem11:
  for t,t1 for xi being Element of dom t st t.xi = [x,s]
  holds dom t c= dom (t with-replacement(xi,t1))
  proof
    let t,t1;
    let xi be Element of dom t;
    assume Z0: t.xi = [x,s];
    let a;
    assume a in dom t;
    then reconsider q = a as Element of dom t;
    xi in Leaves dom t by Z0,Lem13;
    then not xi c< q by TREES_1:def 5;
    then q in dom t with-replacement(xi, dom t1) by TREES_1:def 9;
    hence a in dom (t with-replacement(xi,t1)) by TREES_2:def 11;
  end;
