
theorem Th2:
  for t being Tree, p,q being FinSequence of NAT st p^q in t holds
  t|(p^q) = (t|p)|q
proof
  let t be Tree, p,q be FinSequence of NAT;
  assume
A1: p^q in t;
  let r be FinSequence of NAT;
A2: p in t by A1,TREES_1:21;
  then q in t|p by A1,TREES_1:def 6;
  then
A3: r in (t|p)|q iff q^r in t|p by TREES_1:def 6;
A4: p^q^r = p^(q^r) by FINSEQ_1:32;
  r in t|(p^q) iff p^q^r in t by A1,TREES_1:def 6;
  hence thesis by A2,A4,A3,TREES_1:def 6;
end;
