let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A holds p => (q => (p '&' q)) in TAUT A
let p, q be Element of CQC-WFF A; :: thesis: p => (q => (p '&' q)) in TAUT A
A1: (((p '&' q) 'or' ('not' p)) 'or' ('not' q)) => (('not' q) 'or' ((p '&' q) 'or' ('not' p))) in TAUT A by Th8;
('not' (p '&' q)) => (('not' p) 'or' ('not' q)) in TAUT A by Th17;
then A2: (p '&' q) 'or' (('not' p) 'or' ('not' q)) in TAUT A by Lm1;
((p '&' q) 'or' (('not' p) 'or' ('not' q))) => (((p '&' q) 'or' ('not' p)) 'or' ('not' q)) in TAUT A by Th27;
then ((p '&' q) 'or' ('not' p)) 'or' ('not' q) in TAUT A by A2, CQC_THE1:46;
then ('not' q) 'or' ((p '&' q) 'or' ('not' p)) in TAUT A by A1, CQC_THE1:46;
then A3: ('not' ('not' q)) => ((p '&' q) 'or' ('not' p)) in TAUT A by Lm1;
( q => (((p '&' q) 'or' ('not' p)) => (('not' p) 'or' (p '&' q))) in TAUT A & (q => (((p '&' q) 'or' ('not' p)) => (('not' p) 'or' (p '&' q)))) => ((q => ((p '&' q) 'or' ('not' p))) => (q => (('not' p) 'or' (p '&' q)))) in TAUT A ) by Th8, LUKASI_1:11, LUKASI_1:13;
then A4: (q => ((p '&' q) 'or' ('not' p))) => (q => (('not' p) 'or' (p '&' q))) in TAUT A by CQC_THE1:46;
q => ('not' ('not' q)) in TAUT A by LUKASI_1:27;
then q => ((p '&' q) 'or' ('not' p)) in TAUT A by A3, LUKASI_1:3;
then q => (('not' p) 'or' (p '&' q)) in TAUT A by A4, CQC_THE1:46;
then q => (('not' ('not' p)) => (p '&' q)) in TAUT A by Lm1;
then A5: ('not' ('not' p)) => (q => (p '&' q)) in TAUT A by LUKASI_1:15;
p => ('not' ('not' p)) in TAUT A by LUKASI_1:27;
hence p => (q => (p '&' q)) in TAUT A by A5, LUKASI_1:3; :: thesis: verum