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 Th8:
  G in rng tree_of_subformulae(F) & H is_immediate_constituent_of
  G implies H in rng tree_of_subformulae(F)
proof
  assume that
A1: G in rng tree_of_subformulae(F) and
A2: H is_immediate_constituent_of G;
  consider x being object such that
A3: x in dom tree_of_subformulae(F) and
A4: G = (tree_of_subformulae(F)).x by A1,FUNCT_1:def 3;
  consider t such that
A5: t = x by A3;
  ex n st t^<*n*> in dom tree_of_subformulae(F) & H = ( tree_of_subformulae
  (F)).(t^<*n*>) by A2,A4,A5,Th7;
  hence thesis by FUNCT_1:def 3;
end;
