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;
