let A be QC-alphabet ; :: thesis: for p, q, r being Element of CQC-WFF A st p => q in TAUT A holds

(p '&' r) => (q '&' r) in TAUT A

let p, q, r be Element of CQC-WFF A; :: thesis: ( p => q in TAUT A implies (p '&' r) => (q '&' r) in TAUT A )

A1: (p => q) => ((q => ('not' r)) => (p => ('not' r))) in TAUT A by LUKASI_1:1;

assume p => q in TAUT A ; :: thesis: (p '&' r) => (q '&' r) in TAUT A

then (q => ('not' r)) => (p => ('not' r)) in TAUT A by A1, CQC_THE1:46;

then A2: ('not' (p => ('not' r))) => ('not' (q => ('not' r))) in TAUT A by LUKASI_1:34;

A3: ('not' (q => ('not' r))) => (q '&' r) in TAUT A by Th16;

(p '&' r) => ('not' (p => ('not' r))) in TAUT A by Th15;

then (p '&' r) => ('not' (q => ('not' r))) in TAUT A by A2, LUKASI_1:3;

hence (p '&' r) => (q '&' r) in TAUT A by A3, LUKASI_1:3; :: thesis: verum

(p '&' r) => (q '&' r) in TAUT A

let p, q, r be Element of CQC-WFF A; :: thesis: ( p => q in TAUT A implies (p '&' r) => (q '&' r) in TAUT A )

A1: (p => q) => ((q => ('not' r)) => (p => ('not' r))) in TAUT A by LUKASI_1:1;

assume p => q in TAUT A ; :: thesis: (p '&' r) => (q '&' r) in TAUT A

then (q => ('not' r)) => (p => ('not' r)) in TAUT A by A1, CQC_THE1:46;

then A2: ('not' (p => ('not' r))) => ('not' (q => ('not' r))) in TAUT A by LUKASI_1:34;

A3: ('not' (q => ('not' r))) => (q '&' r) in TAUT A by Th16;

(p '&' r) => ('not' (p => ('not' r))) in TAUT A by Th15;

then (p '&' r) => ('not' (q => ('not' r))) in TAUT A by A2, LUKASI_1:3;

hence (p '&' r) => (q '&' r) in TAUT A by A3, LUKASI_1:3; :: thesis: verum