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
  rng tree_of_subformulae(F) = Subformulae(F)
proof
  thus rng tree_of_subformulae(F) c= Subformulae(F)
  proof
    let y be object;
    assume
A1: y in rng tree_of_subformulae(F);
    then y is Element of QC-WFF(A) by RELAT_1:167;
    then consider G such that
A2: G = y;
    G is_subformula_of F by A1,A2,Th10;
    hence thesis by A2,QC_LANG2:def 22;
  end;
  thus Subformulae(F) c= rng tree_of_subformulae(F)
  proof
    let y be object;
    assume y in Subformulae(F);
    then ex G st G = y & G is_subformula_of F by QC_LANG2:def 22;
    hence thesis by Th10;
  end;
end;
