let Al be QC-alphabet ; :: thesis: for f, g being FinSequence of CQC-WFF Al st Ant f = Ant g & Ant f |= Suc f & Ant g |= Suc g holds

Ant f |= (Suc f) '&' (Suc g)

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

assume A1: ( Ant f = Ant g & Ant f |= Suc f & Ant g |= Suc g ) ; :: thesis: Ant f |= (Suc f) '&' (Suc g)

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 |= (Suc f) '&' (Suc g)

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 |= (Suc f) '&' (Suc g)

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

assume J,v |= Ant f ; :: thesis: J,v |= (Suc f) '&' (Suc g)

then ( J,v |= Suc f & J,v |= Suc g ) by A1;

hence J,v |= (Suc f) '&' (Suc g) by VALUAT_1:18; :: thesis: verum

Ant f |= (Suc f) '&' (Suc g)

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

assume A1: ( Ant f = Ant g & Ant f |= Suc f & Ant g |= Suc g ) ; :: thesis: Ant f |= (Suc f) '&' (Suc g)

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 |= (Suc f) '&' (Suc g)

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 |= (Suc f) '&' (Suc g)

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

assume J,v |= Ant f ; :: thesis: J,v |= (Suc f) '&' (Suc g)

then ( J,v |= Suc f & J,v |= Suc g ) by A1;

hence J,v |= (Suc f) '&' (Suc g) by VALUAT_1:18; :: thesis: verum