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;
reserve G1, G2 for Subformula of F,
  t1 for Entry_Point_in_Subformula_Tree of G1,
  t2 for Entry_Point_in_Subformula_Tree of G2;

theorem
  (ex t1,t2 st t1 is_a_prefix_of t2) implies G2 is_subformula_of G1
proof
  given t1,t2 such that
A1: t1 is_a_prefix_of t2;
  (tree_of_subformulae(F)).t1 = G1 & (tree_of_subformulae(F)).t2 = G2 by Def5;
  hence thesis by A1,Th13;
end;
