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 => (p => r => (p => (q '&' r))) is tautology
  proof
    let M;
    thus (SAT M).(p => q => (p => r => (p => (q '&' r))))
= (SAT M).(p =>q) => (SAT M).(p => r => (p => (q '&' r))) by Def11
.= (SAT M).p => (SAT M).q => ((SAT M).(p => r => (p => (q '&' r)))) by Def11
.= (SAT M).p => (SAT M).q
=> ((SAT M).(p => r) => (SAT M).(p => (q '&' r))) by Def11
.= (SAT M).p => (SAT M).q =>
((SAT M).p => (SAT M).r => (SAT M).(p => (q '&' r))) by Def11
.= (SAT M).p => (SAT M).q =>
((SAT M).p => (SAT M).r => ((SAT M).p => (SAT M).(q '&' r))) by Def11
.= (SAT M).p => (SAT M).q =>
((SAT M).p => (SAT M).r => ((SAT M).p => ((SAT M).q '&' (SAT M).r))) by semcon2
.= 1 by th5;
end;
