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
  G2 is_subformula_of G1 implies for t1 ex t2 st t1 is_a_prefix_of t2
proof
  assume
A1: G2 is_subformula_of G1;
  now
    let t1;
    consider H being Element of QC-WFF(A) such that
A2: H = G2;
    reconsider H as Subformula of G1 by A1,A2,Def4;
    set s = the Entry_Point_in_Subformula_Tree of H;
    (tree_of_subformulae(G1)).s = H by Def5;
    then s in G1-entry_points_in_subformula_tree_of G2 by A2,Def3;
    then t1^s is Entry_Point_in_Subformula_Tree of G2 by Th37;
    then consider t2 such that
A3: t2 = t1^s;
    take t2;
    thus t1 is_a_prefix_of t2 by A3,TREES_1:1;
  end;
  hence thesis;
end;
