let Al be QC-alphabet ; :: thesis: for p being Element of CQC-WFF Al
for f, g being FinSequence of CQC-WFF Al st len f > 1 & Ant f = Ant g & 'not' p = Suc (Ant f) & 'not' (Suc f) = Suc g & Ant f |= Suc f & Ant g |= Suc g holds
Ant (Ant f) |= p
let p be Element of CQC-WFF Al; :: thesis: for f, g being FinSequence of CQC-WFF Al st len f > 1 & Ant f = Ant g & 'not' p = Suc (Ant f) & 'not' (Suc f) = Suc g & Ant f |= Suc f & Ant g |= Suc g holds
Ant (Ant f) |= p
let f, g be FinSequence of CQC-WFF Al; :: thesis: ( len f > 1 & Ant f = Ant g & 'not' p = Suc (Ant f) & 'not' (Suc f) = Suc g & Ant f |= Suc f & Ant g |= Suc g implies Ant (Ant f) |= p )
assume that
A1: len f > 1 and
A2: Ant f = Ant g and
A3: 'not' p = Suc (Ant f) and
A4: ( 'not' (Suc f) = Suc g & Ant f |= Suc f & Ant g |= Suc g ) ; :: thesis: Ant (Ant f) |= p
A5: len (Ant f) > 0 by A1, Th4;

A6: now :: thesis: for A being non empty set
for J being interpretation of Al,A
for v being Element of Valuations_in (Al,A) holds not J,v |= Ant f

given A being non empty set , J being interpretation of Al,A, v being Element of Valuations_in (Al,A) such that A7: J,v |= Ant f ; :: thesis: contradiction
( J,v |= Suc f & J,v |= 'not' (Suc f) ) by A2, A4, A7, Def15;
hence contradiction by VALUAT_1:17; :: thesis: verum

end;

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 (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 (Ant f) holds
J,v |= p
let v be Element of Valuations_in (Al,A); :: thesis: ( J,v |= Ant (Ant f) implies J,v |= p )
assume A8: J,v |= Ant (Ant f) ; :: thesis: J,v |= p

now :: thesis: not J,v |= Suc (Ant f)

assume J,v |= Suc (Ant f) ; :: thesis: contradiction
then J,v |= Ant f by A5, A8, Th17;
hence contradiction by A6; :: thesis: verum

end;

hence J,v |= p by A3, VALUAT_1:17; :: thesis: verum