reserve A for QC-alphabet;
reserve n,k,m for Nat;
reserve F,G,G9,H,H9 for Element of QC-WFF(A);
reserve t, t9, t99 for Element of dom tree_of_subformulae(F);

theorem Th6:
  t^<*n*> in dom tree_of_subformulae(F) implies ex G st G = (
  tree_of_subformulae(F)).(t^<*n*>) & G is_immediate_constituent_of (
  tree_of_subformulae(F)).t
proof
A1: succ t = {t^<*k*>: t^<*k*> in dom tree_of_subformulae(F)};
  assume t^<*n*> in dom tree_of_subformulae(F);
  then consider t9 such that
A2: t9 = t^<*n*>;
A3: rng list_of_immediate_constituents((tree_of_subformulae(F)).t) = { G1
  where G1 is Element of QC-WFF(A) : G1 is_immediate_constituent_of (
  tree_of_subformulae(F)).t } by Th4;
  consider G such that
A4: G = (tree_of_subformulae(F)).t9;
  t9 in {t^<*k*>: t^<*k*> in dom tree_of_subformulae(F)} by A2;
  then G in rng succ((tree_of_subformulae(F)),t) by A4,A1,TREES_9:41;
  then G in rng list_of_immediate_constituents((tree_of_subformulae(F)).t) by
Def2;
  hence thesis by A2,A4,A3;
end;
