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

Ant g |= Suc g

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

assume that

A1: Ant f is_Subsequence_of Ant g and

A2: ( Suc f = Suc g & Ant f |= Suc f ) ; :: thesis: Ant g |= 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 g holds

J,v |= Suc g

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

J,v |= Suc g

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

assume A3: J,v |= rng (Ant g) ; :: according to CALCUL_1:def 14 :: thesis: J,v |= Suc g

then J,v |= Ant f ;

hence J,v |= Suc g by A2; :: thesis: verum

Ant g |= Suc g

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

assume that

A1: Ant f is_Subsequence_of Ant g and

A2: ( Suc f = Suc g & Ant f |= Suc f ) ; :: thesis: Ant g |= 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 g holds

J,v |= Suc g

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

J,v |= Suc g

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

assume A3: J,v |= rng (Ant g) ; :: according to CALCUL_1:def 14 :: thesis: J,v |= Suc g

now :: thesis: for p being Element of CQC-WFF Al st p in rng (Ant f) holds

J,v |= p

then
J,v |= rng (Ant f)
;J,v |= p

let p be Element of CQC-WFF Al; :: thesis: ( p in rng (Ant f) implies J,v |= p )

assume A4: p in rng (Ant f) ; :: thesis: J,v |= p

rng (Ant f) c= rng (Ant g) by A1, Th1;

hence J,v |= p by A3, A4; :: thesis: verum

end;assume A4: p in rng (Ant f) ; :: thesis: J,v |= p

rng (Ant f) c= rng (Ant g) by A1, Th1;

hence J,v |= p by A3, A4; :: thesis: verum

then J,v |= Ant f ;

hence J,v |= Suc g by A2; :: thesis: verum