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
  p in Leaves Subformulae q implies p = VERUM or p is simple
proof
  assume p in Leaves Subformulae q;
  then p in (Subformulae q).:Leaves dom Subformulae q by TREES_2:def 9;
  then consider x being object such that
A1: x in dom Subformulae q and
A2: x in Leaves dom Subformulae q and
A3: p = (Subformulae q).x by FUNCT_1:def 6;
  reconsider f = x as Element of dom Subformulae q by A1;
A4: (Subformulae q)|f = Subformulae p by A3,Th35;
A5: p is not conditional
  proof
    assume not thesis;
    then consider r,s such that
A6: p = r => s;
    Subformulae p = p-tree(Subformulae r,Subformulae s) by A6,Th33;
    hence contradiction by A2,A4,TREES_9:6;
  end;
  p is not conjunctive
  proof
    assume not thesis;
    then consider r,s such that
A7: p = r '&' s;
    Subformulae p = p-tree(Subformulae r,Subformulae s) by A7,Th32;
    hence contradiction by A2,A4,TREES_9:6;
  end;
  hence thesis by A5,Th9;
end;
