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.1 = 3+n implies p is simple
proof
  assume
A1: p.1 = 3+n;
  per cases by Th9;
  suppose
    p is conjunctive;
    then consider r,s such that
A2: p = r '&' s;
    p = <*2*>^(r^s) by A2,FINSEQ_1:32;
    then 2+0 = 2+(1+n) by A1,FINSEQ_1:41;
    hence thesis;
  end;
  suppose
    p is conditional;
    then consider r,s such that
A3: p = r => s;
    p = <*1*>^(r^s) by A3,FINSEQ_1:32;
    then 1+0 = 1+(2+n) by A1,FINSEQ_1:41;
    hence thesis;
  end;
  suppose
    p is simple;
    hence thesis;
  end;
  suppose
    p = VERUM;
    hence thesis by A1;
  end;
end;
