reserve p,q,r,s,A,B for Element of PL-WFF,
  F,G,H for Subset of PL-WFF,
  k,n for Element of NAT,
  f,f1,f2 for FinSequence of PL-WFF;
reserve M for PLModel;

theorem
  ('not' 'not' p) <=> p is tautology
  proof
    let M;
    thus (SAT M).(('not' 'not' p) <=> p) =
    (SAT M).('not' 'not' p) <=> (SAT M).p by semequ2
    .= ('not' (SAT M).('not' p)) <=> (SAT M).p by semnot2
    .= ('not' 'not' (SAT M).p) <=> (SAT M).p by semnot2
    .=1 by th2;
  end;
