theorem
  for vt being finite DecoratedTree st vt is_an_evaluation_of t,f for p
  being Node of t holds vt.p = (t|p)@f
proof
  let vt be finite DecoratedTree such that
A1: vt is_an_evaluation_of t,f;
  let p be Node of t;
  reconsider n = p as Node of vt by A1;
A2: n^{} = n by FINSEQ_1:34;
A3: {} in (dom vt)|p by TREES_1:22;
  (t|p)@f = (vt|n).<*>NAT by A1,Th34,Th39;
  hence thesis by A2,A3,TREES_2:def 10;
end;
