reserve X,x for set;
reserve k,m,n for Element of NAT,
  p,q,r,s,r9,s9 for Element of HP-WFF,
  T1,T2 for Tree;
reserve T1,T2 for DecoratedTree;
reserve t,t1 for FinSequence;

theorem Th34:
  (Subformulae p).{} = p
proof
  per cases by Th9;
  suppose
    p is conjunctive;
    then consider r,s such that
A1: p = r '&' s;
    Subformulae p = p-tree(Subformulae r,Subformulae s) by A1,Th32;
    hence thesis by Th6;
  end;
  suppose
    p is conditional;
    then consider r,s such that
A2: p = r => s;
    Subformulae p = p-tree(Subformulae r,Subformulae s) by A2,Th33;
    hence thesis by Th6;
  end;
  suppose
    p is simple;
    then ex n st p = prop n;
    then Subformulae p = root-tree p by Def9;
    hence thesis by TREES_4:3;
  end;
  suppose
    p = VERUM;
    hence thesis by Th30,TREES_4:3;
  end;
end;
