theorem Th130:
  for t,t1 being DecoratedTree, xi being Node of t holds
  (t with-replacement(xi,t1))|xi = t1
  proof
    let t,t1 be DecoratedTree;
    let xi be Node of t;
A1: xi in dom t with-replacement(xi,dom t1)
    = dom(t with-replacement(xi,t1)) by TREES_1:def 9,TREES_2:def 11;
A2: dom((t with-replacement(xi,t1))|xi)
    = (dom (t with-replacement(xi,t1)))|xi by TREES_2:def 10;
    hence dom((t with-replacement(xi,t1))|xi) = dom t1 by A1,Th129;
    let p be Node of (t with-replacement(xi,t1))|xi;
    xi^p in dom(t with-replacement(xi,t1)) & xi c= xi^p
    by A1,A2,TREES_1:1,def 6;
    then consider r being FinSequence of NAT such that
A4: r in dom t1 & xi^p = xi^r & (t with-replacement(xi,t1)).(xi^p) = t1.r
    by A1,TREES_2:def 11;
    thus ((t with-replacement(xi,t1))|xi).p
    = (t with-replacement(xi,t1)).(xi^p) by A2,TREES_2:def 10
    .= t1.p by A4,FINSEQ_1:33;
  end;
