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' (p '&' q) <=> ('not' p) 'or' ('not' q) is tautology
 proof
   let M;
   thus (SAT M).('not' (p '&' q) <=> ('not' p) 'or' ('not' q)) =
   (SAT M).('not' (p '&' q)) <=> (SAT M).(('not' p) 'or' ('not' q)) by semequ2
   .= ('not' (SAT M).(p '&' q)) <=>
   (SAT M).(('not' p) 'or' ('not' q)) by semnot2
   .= ('not' ((SAT M).p '&' (SAT M).q)) <=>
   (SAT M).(('not' p) 'or' ('not' q)) by semcon2
   .= ('not' ((SAT M).p '&' (SAT M).q)) <=>
   ((SAT M).('not' p) 'or' (SAT M).('not' q)) by semdis2
   .= ('not' ((SAT M).p '&' (SAT M).q)) <=>
   ('not' (SAT M).p 'or' (SAT M).('not' q)) by semnot2
   .= ('not' ((SAT M).p '&' (SAT M).q)) <=>
   ('not' (SAT M).p 'or' 'not' (SAT M).q) by semnot2
   .= 1 by th3;
end;
