let Al be QC-alphabet ; :: thesis: for f being FinSequence of CQC-WFF Al st Suc f is_tail_of Ant f holds

Ant f |= Suc f

let f be FinSequence of CQC-WFF Al; :: thesis: ( Suc f is_tail_of Ant f implies Ant f |= Suc f )

assume Suc f is_tail_of Ant f ; :: thesis: Ant f |= Suc f

then ex i being Nat st

( i in dom (Ant f) & (Ant f) . i = Suc f ) by Lm1;

then A1: Suc f in rng (Ant f) by FUNCT_1:3;

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

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

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

assume J,v |= rng (Ant f) ; :: according to CALCUL_1:def 14 :: thesis: J,v |= Suc f

hence J,v |= Suc f by A1; :: thesis: verum

Ant f |= Suc f

let f be FinSequence of CQC-WFF Al; :: thesis: ( Suc f is_tail_of Ant f implies Ant f |= Suc f )

assume Suc f is_tail_of Ant f ; :: thesis: Ant f |= Suc f

then ex i being Nat st

( i in dom (Ant f) & (Ant f) . i = Suc f ) by Lm1;

then A1: Suc f in rng (Ant f) by FUNCT_1:3;

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

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

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

assume J,v |= rng (Ant f) ; :: according to CALCUL_1:def 14 :: thesis: J,v |= Suc f

hence J,v |= Suc f by A1; :: thesis: verum