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;

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

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

let A be non empty set ; :: according to CALCUL_1:def 15 :: thesis: for J being interpretation of Al,Afor 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;

hence contradiction by VALUAT_1:17; :: thesis: verum

end;( J,v |= Suc f & J,v |= 'not' (Suc f) ) by A2, A4, A7;

hence contradiction by VALUAT_1:17; :: thesis: verum

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)

hence
J,v |= p
by A3, VALUAT_1:17; :: thesis: verumassume
J,v |= Suc (Ant f)
; :: thesis: contradiction

then J,v |= Ant f by A5, A8, Th17;

hence contradiction by A6; :: thesis: verum

end;then J,v |= Ant f by A5, A8, Th17;

hence contradiction by A6; :: thesis: verum