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 Th10:
  len p >= 1
proof
  per cases by Th9;
  suppose
    p is conjunctive;
    then consider r,s such that
A1: p = r '&' s;
    len p = len(<*2*>^(r^s)) by A1,FINSEQ_1:32
      .= len<*2*> + len(r^s) by FINSEQ_1:22
      .= 1 + len(r^s) by FINSEQ_1:39;
    hence thesis by NAT_1:11;
  end;
  suppose
    p is conditional;
    then consider r,s such that
A2: p = r => s;
    len p = len(<*1*>^(r^s)) by A2,FINSEQ_1:32
      .= len<*1*> + len(r^s) by FINSEQ_1:22
      .= 1 + len(r^s) by FINSEQ_1:39;
    hence thesis by NAT_1:11;
  end;
  suppose
    p is simple;
    then ex n st p = prop n;
    hence thesis by FINSEQ_1:39;
  end;
  suppose
    p = VERUM;
    hence thesis by FINSEQ_1:39;
  end;
end;
