reserve i,j,x,y for object,
  f,g for Function;
reserve T,T1 for finite Tree,
  t,p for Element of T,
  t1 for Element of T1;

theorem Th11:
  for p being Tree-yielding FinSequence, k being Element of NAT st
    k+1 in dom p holds (tree p)|<*k*> = p.(k+1)
proof
  let p be Tree-yielding FinSequence, k be Element of NAT;
  assume k+1 in dom p;
  then k+1 <= len p by FINSEQ_3:25;
  then k < len p by NAT_1:13;
  hence thesis by TREES_3:49;
end;
