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 Th7:
  H is_immediate_constituent_of (tree_of_subformulae(F)).t iff ex
n st t^<*n*> in dom tree_of_subformulae(F) & H = (tree_of_subformulae(F)).(t^<*
  n*>)
proof
  now
    set G = (tree_of_subformulae(F)).t;
    assume H is_immediate_constituent_of (tree_of_subformulae(F)).t;
    then H in {H1 where H1 is Element of QC-WFF(A) : H1
    is_immediate_constituent_of G };
    then
A1: H in rng list_of_immediate_constituents(G) by Th4;
    succ(tree_of_subformulae(F),t) = list_of_immediate_constituents(G) by Def2;
    then consider t9 such that
A2: H = (tree_of_subformulae(F)).t9 and
A3: t9 in succ t by A1,TREES_9:42;
    ex n st t9 = t^<*n*> & t^<*n*> in dom tree_of_subformulae(F) by A3;
    hence ex n st t^<*n*> in dom tree_of_subformulae(F) & H = (
    tree_of_subformulae(F)).(t^<*n*>) by A2;
  end;
  hence H is_immediate_constituent_of (tree_of_subformulae(F)).t implies ex n
st t^<*n*> in dom tree_of_subformulae(F) & H = (tree_of_subformulae(F)).(t^<*n
  *>);
  given n such that
A4: t^<*n*> in dom tree_of_subformulae(F) and
A5: H = (tree_of_subformulae(F)).(t^<*n*>);
  ex G st G = (tree_of_subformulae(F)).(t^<*n*>) & G
  is_immediate_constituent_of (tree_of_subformulae(F)).t by A4,Th6;
  hence thesis by A5;
end;
