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 Th9:
  p is conjunctive or p is conditional or p is simple or p = VERUM
proof
  defpred P[Element of HP-WFF] means $1 is conjunctive or $1 is conditional or
  $1 is simple or $1 = VERUM;
A1: P[VERUM];
A2: for n holds P[prop n] by Def8;
A3: for r,s st P[r] & P[s] holds P[r '&' s] & P[r => s] by Def6,Def7;
  for p holds P[p] from HPInd(A1,A2,A3);
  hence thesis;
end;
