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
  (p '&' q '&' r) <=> (p '&' (q '&' r)) is tautology
  proof
    let M;
    thus (SAT M).((p '&' q '&' r) <=> (p '&' (q '&' r)))
    = (SAT M).(p '&' q '&' r) <=> (SAT M).(p '&' (q '&' r)) by semequ2
    .= (SAT M).(p '&' q) '&' (SAT M).r <=> (SAT M).(p '&' (q '&' r))
    by semcon2
    .= (SAT M).(p '&' q) '&' (SAT M).r
    <=> ((SAT M).p '&' (SAT M).(q '&' r)) by semcon2
    .= (SAT M).(p '&' q) '&' (SAT M).r
    <=> ((SAT M).p '&' ((SAT M).q '&' (SAT M).r)) by semcon2
    .= (SAT M).p '&' (SAT M).q '&' (SAT M).r
    <=> ((SAT M).p '&' ((SAT M).q '&' (SAT M).r)) by semcon2
    .= 1 by th7;
end;
