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);
reserve x for set;
reserve x,y for set;
reserve t for Element of dom tree_of_subformulae(F),
  s for Element of dom tree_of_subformulae(G);
reserve t for Element of dom tree_of_subformulae(F),
  s for FinSequence;
reserve C for Chain of dom tree_of_subformulae(F);
reserve G for Subformula of F;
reserve t, t9 for Entry_Point_in_Subformula_Tree of G;
reserve G1, G2 for Subformula of F,
  t1 for Entry_Point_in_Subformula_Tree of G1,
  s for Element of dom tree_of_subformulae(G1);
reserve s for FinSequence;

theorem
  t1^s is Entry_Point_in_Subformula_Tree of G2 implies s in G1
  -entry_points_in_subformula_tree_of G2
proof
  consider t9 being FinSequence such that
A1: t9 = t1^s;
  (tree_of_subformulae(F)).t1 = G1 by Def5;
  then
A2: t1 in F-entry_points_in_subformula_tree_of G1 by Def3;
  assume t1^s is Entry_Point_in_Subformula_Tree of G2;
  then
  t9 in { t2 where t2 is Entry_Point_in_Subformula_Tree of G2 : t2 = t2 }
  by A1;
  then t9 in entry_points_in_subformula_tree(G2) by Th36;
  hence thesis by A2,A1,Th28;
end;
