let Al be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF Al

for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds

Ant f |= p

let p, q be Element of CQC-WFF Al; :: thesis: for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds

Ant f |= p

let f be FinSequence of CQC-WFF Al; :: thesis: ( Ant f |= p '&' q implies Ant f |= p )

assume A1: Ant f |= p '&' q ; :: thesis: Ant f |= p

let A be non empty set ; :: according to CALCUL_1:def 15 :: thesis: for J being interpretation of Al,A

for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds

J,v |= p

let J be interpretation of Al,A; :: thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds

J,v |= p

let v be Element of Valuations_in (Al,A); :: thesis: ( J,v |= Ant f implies J,v |= p )

assume J,v |= Ant f ; :: thesis: J,v |= p

then J,v |= p '&' q by A1;

hence J,v |= p by VALUAT_1:18; :: thesis: verum

for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds

Ant f |= p

let p, q be Element of CQC-WFF Al; :: thesis: for f being FinSequence of CQC-WFF Al st Ant f |= p '&' q holds

Ant f |= p

let f be FinSequence of CQC-WFF Al; :: thesis: ( Ant f |= p '&' q implies Ant f |= p )

assume A1: Ant f |= p '&' q ; :: thesis: Ant f |= p

let A be non empty set ; :: according to CALCUL_1:def 15 :: thesis: for J being interpretation of Al,A

for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds

J,v |= p

let J be interpretation of Al,A; :: thesis: for v being Element of Valuations_in (Al,A) st J,v |= Ant f holds

J,v |= p

let v be Element of Valuations_in (Al,A); :: thesis: ( J,v |= Ant f implies J,v |= p )

assume J,v |= Ant f ; :: thesis: J,v |= p

then J,v |= p '&' q by A1;

hence J,v |= p by VALUAT_1:18; :: thesis: verum