reserve A for QC-alphabet;
reserve p, q, r, s for Element of CQC-WFF(A);

theorem
  ( p '&' ( q 'or' r )) => (( p '&' q ) 'or' ( p '&' r )) in TAUT(A)
proof
A1: 'not' (( p '&' q ) 'or' ( p '&' r )) => ( 'not' ( p '&' q ) '&' 'not' (
  p '&' r )) in TAUT(A) by Th6;
  'not' ( p => 'not' q ) => ( p '&' q ) in TAUT(A) & ('not' ( p => 'not' q )
  => ( p '&' q )) => ( 'not' ( p '&' q ) => ( p => 'not' q )) in TAUT(A)
by Th16,LUKASI_1:31;
  then
A2: 'not' ( p '&' q ) => ( p => 'not' q ) in TAUT(A) by CQC_THE1:46;
  'not' ( p => 'not' r ) => ( p '&' r ) in TAUT(A) & ('not' ( p => 'not' r )
  => ( p '&' r )) => ( 'not' ( p '&' r ) => ( p => 'not' r )) in TAUT(A)
by Th16,LUKASI_1:31;
  then
A3: 'not' ( p '&' r ) => ( p => 'not' r ) in TAUT(A) by CQC_THE1:46;
  ( p => 'not' q ) => (( p => 'not' r ) => ( p => ( 'not' q '&' 'not' r ))
) in TAUT(A) & ( p => ( 'not' q '&' 'not' r )) =
  'not' ( p '&' 'not' ( 'not' q '&'
  'not' r )) by Th33,QC_LANG2:def 2;
  then
  ( p => 'not' q ) => (( p => 'not' r ) => 'not' ( p '&' ( q 'or' r ))) in
  TAUT(A) by QC_LANG2:def 3;
  then 'not' ( p '&' q ) => (( p => 'not' r ) => 'not' ( p '&' ( q 'or' r )))
  in TAUT(A) by A2,LUKASI_1:3;
  then
  ( p => 'not' r ) => ( 'not' ( p '&' q ) => 'not' ( p '&' ( q 'or' r )))
  in TAUT(A) by LUKASI_1:15;
  then
  'not' ( p '&' r ) => ('not' ( p '&' q ) => 'not' ( p '&' ( q 'or' r )))
  in TAUT(A) by A3,LUKASI_1:3;
  then
A4: 'not' ( p '&' q ) => ('not' ( p '&' r ) => 'not' ( p '&' ( q 'or' r )))
  in TAUT(A) by LUKASI_1:15;
  ( 'not' ( p '&' q ) => ('not' ( p '&' r ) => 'not' ( p '&' ( q 'or' r )
))) => ((( 'not' ( p '&' q ) '&' 'not' ( p '&' r )) => 'not' ( p '&' ( q 'or' r
  )))) in TAUT(A) by Th32;
  then ( 'not' ( p '&' q ) '&' 'not' ( p '&' r )) => 'not' ( p '&' ( q 'or' r
  )) in TAUT(A) by A4,CQC_THE1:46;
  then 'not' (( p '&' q ) 'or' ( p '&' r )) => 'not' ( p '&' ( q 'or' r )) in
  TAUT(A) by A1,LUKASI_1:3;
  hence thesis by LUKASI_1:35;
end;
