let p, q be Element of CQC-WFF ; :: thesis: (p => (q '&' ('not' q))) => ('not' p) in TAUT
p => ('not' (q '&' ('not' q))) in TAUT by Th1, LUKASI_1:13;
then A1: ('not' ('not' (q '&' ('not' q)))) => ('not' p) in TAUT by LUKASI_1:34;
(q '&' ('not' q)) => ('not' ('not' (q '&' ('not' q)))) in TAUT by LUKASI_1:27;
then (q '&' ('not' q)) => ('not' p) in TAUT by A1, LUKASI_1:3;
then A2: p => ((q '&' ('not' q)) => ('not' p)) in TAUT by LUKASI_1:13;
(p => ((q '&' ('not' q)) => ('not' p))) => ((p => (q '&' ('not' q))) => (p => ('not' p))) in TAUT by LUKASI_1:11;
then A3: (p => (q '&' ('not' q))) => (p => ('not' p)) in TAUT by A2, CQC_THE1:82;
A4: (('not' ('not' p)) => ('not' p)) => ('not' p) in TAUT by CQC_THE1:78;
A5: ('not' ('not' p)) => p in TAUT by LUKASI_1:25;
(('not' ('not' p)) => p) => ((p => ('not' p)) => (('not' ('not' p)) => ('not' p))) in TAUT by LUKASI_1:1;
then (p => ('not' p)) => (('not' ('not' p)) => ('not' p)) in TAUT by A5, CQC_THE1:82;
then (p => ('not' p)) => ('not' p) in TAUT by A4, LUKASI_1:3;
hence (p => (q '&' ('not' q))) => ('not' p) in TAUT by A3, LUKASI_1:3; :: thesis: verum