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;

theorem Th30:
  (tree_of_subformulae(F))|t = tree_of_subformulae(( tree_of_subformulae(F)).t)
proof
  set T1 = (tree_of_subformulae(F))|t;
  set T2 = tree_of_subformulae((tree_of_subformulae(F)).t);
  thus
A1: dom T1 = dom T2
  proof
    let p be FinSequence of NAT;
    now
      consider G such that
A2:   G = (tree_of_subformulae(F)).t;
A3:   t in F-entry_points_in_subformula_tree_of G by A2,Def3;
      consider t9 being FinSequence of NAT such that
A4:   t9 = t^p;
      assume p in dom T1;
      then p in (dom tree_of_subformulae(F))|t by TREES_2:def 10;
      then reconsider t9 as Element of dom tree_of_subformulae(F) by A4,
TREES_1:def 6;
      consider H such that
A5:   H = (tree_of_subformulae(F)).t9;
A6:   G-entry_points_in_subformula_tree_of H c= dom T2 by A2,TREES_1:def 11;
      t9 in F-entry_points_in_subformula_tree_of H by A5,Def3;
      then p in G-entry_points_in_subformula_tree_of H by A3,A4,Th28;
      hence p in dom T2 by A6;
    end;
    hence p in dom T1 implies p in dom T2;
    now
      consider G such that
A7:   G = (tree_of_subformulae(F)).t;
      assume p in dom T2;
      then reconsider s = p as Element of dom tree_of_subformulae(G) by A7;
      consider H such that
A8:   H = (tree_of_subformulae(G)).s;
A9:   s in G-entry_points_in_subformula_tree_of H by A8,Def3;
A10:  F-entry_points_in_subformula_tree_of H c= dom tree_of_subformulae(F
      ) by TREES_1:def 11;
      t in F-entry_points_in_subformula_tree_of G by A7,Def3;
      then t^s in F-entry_points_in_subformula_tree_of H by A9,Th27;
      then s in (dom tree_of_subformulae(F))|t by A10,TREES_1:def 6;
      hence p in dom T1 by TREES_2:def 10;
    end;
    hence thesis;
  end;
  now
    let p be Node of T1;
    consider G such that
A11: G = (tree_of_subformulae(F)).t;
    reconsider s = p as Element of dom tree_of_subformulae(G) by A1,A11;
A12: dom T1 = (dom tree_of_subformulae(F))|t by TREES_2:def 10;
    then reconsider t9= t^s as Element of dom tree_of_subformulae(F) by
TREES_1:def 6;
    consider H such that
A13: H = T1.p;
A14: t in F-entry_points_in_subformula_tree_of G by A11,Def3;
    T1.p = (tree_of_subformulae(F)).(t^p) by A12,TREES_2:def 10;
    then t9 in F-entry_points_in_subformula_tree_of H by A13,Def3;
    then s in G-entry_points_in_subformula_tree_of H by A14,Th28;
    hence T1.p = T2.p by A11,A13,Def3;
  end;
  hence for p being Node of T1 holds T1.p = T2.p;
end;
