let Al be QC-alphabet ; 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; ( 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 )
; Ant g |= Suc g
let A be non empty set ; CALCUL_1:def 15 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; 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); ( J,v |= Ant g implies J,v |= Suc g )
assume A3:
J,v |= rng (Ant g)
; CALCUL_1:def 14 J,v |= Suc g
then
J,v |= rng (Ant f)
;
then
J,v |= Ant f
;
hence
J,v |= Suc g
by A2; verum