let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A holds ('not' (p => ('not' q))) => (p '&' q) in TAUT A
let p, q be Element of CQC-WFF A; :: thesis: ('not' (p => ('not' q))) => (p '&' q) in TAUT A
A1: ('not' ('not' (p '&' ('not' ('not' q))))) => (p '&' ('not' ('not' q))) in TAUT A by LUKASI_1:25;
( (p '&' ('not' ('not' q))) => (('not' ('not' q)) '&' p) in TAUT A & (('not' ('not' q)) '&' p) => (q '&' p) in TAUT A ) by Lm3, CQC_THE1:45;
then A2: (p '&' ('not' ('not' q))) => (q '&' p) in TAUT A by LUKASI_1:3;
(q '&' p) => (p '&' q) in TAUT A by CQC_THE1:45;
then (p '&' ('not' ('not' q))) => (p '&' q) in TAUT A by A2, LUKASI_1:3;
then ('not' ('not' (p '&' ('not' ('not' q))))) => (p '&' q) in TAUT A by A1, LUKASI_1:3;
hence ('not' (p => ('not' q))) => (p '&' q) in TAUT A by QC_LANG2:def 2; :: thesis: verum