:: CALCUL_1 semantic presentation
:: deftheorem Def1 defines Ant CALCUL_1:def 1 :
:: deftheorem Def2 defines Suc CALCUL_1:def 2 :
:: deftheorem Def3 defines is_tail_of CALCUL_1:def 3 :
:: deftheorem Def4 defines is_Subsequence_of CALCUL_1:def 4 :
theorem Th1: :: CALCUL_1:1
theorem Th2: :: CALCUL_1:2
theorem Th3: :: CALCUL_1:3
theorem Th4: :: CALCUL_1:4
theorem Th5: :: CALCUL_1:5
theorem Th6: :: CALCUL_1:6
theorem Th7: :: CALCUL_1:7
theorem Th8: :: CALCUL_1:8
theorem Th9: :: CALCUL_1:9
theorem Th10: :: CALCUL_1:10
theorem Th11: :: CALCUL_1:11
theorem Th12: :: CALCUL_1:12
theorem Th13: :: CALCUL_1:13
theorem Th14: :: CALCUL_1:14
:: deftheorem Def5 defines still_not-bound_in CALCUL_1:def 5 :
:: deftheorem Def6 defines set_of_CQC-WFF-seq CALCUL_1:def 6 :
definition
let c
1 be
FinSequence of
[:set_of_CQC-WFF-seq ,Proof_Step_Kinds :];
let c
2 be
Nat;
pred c
1,c
2 is_a_correct_step means :
Def7:
:: CALCUL_1:def 7
ex b
1 being
FinSequence of
CQC-WFF st
(
Suc b
1 is_tail_of Ant b
1 &
(a1 . a2) `1 = b
1 )
if (a1 . a2) `2 = 0
ex b
1 being
FinSequence of
CQC-WFF st
(a1 . a2) `1 = b
1 ^ <*VERUM *> if (a1 . a2) `2 = 1
ex b
1 being
Natex b
2, b
3 being
FinSequence of
CQC-WFF st
( 1
<= b
1 & b
1 < a
2 &
Ant b
2 is_Subsequence_of Ant b
3 &
Suc b
2 = Suc b
3 &
(a1 . b1) `1 = b
2 &
(a1 . a2) `1 = b
3 )
if (a1 . a2) `2 = 2
ex b
1, b
2 being
Natex b
3, b
4 being
FinSequence of
CQC-WFF st
( 1
<= b
1 & b
1 < a
2 & 1
<= b
2 & b
2 < b
1 &
len b
3 > 1 &
len b
4 > 1 &
Ant (Ant b3) = Ant (Ant b4) &
'not' (Suc (Ant b3)) = Suc (Ant b4) &
Suc b
3 = Suc b
4 & b
3 = (a1 . b2) `1 & b
4 = (a1 . b1) `1 &
(Ant (Ant b3)) ^ <*(Suc b3)*> = (a1 . a2) `1 )
if (a1 . a2) `2 = 3
ex b
1, b
2 being
Natex b
3, b
4 being
FinSequence of
CQC-WFF ex b
5 being
Element of
CQC-WFF st
( 1
<= b
1 & b
1 < a
2 & 1
<= b
2 & b
2 < b
1 &
len b
3 > 1 &
Ant b
3 = Ant b
4 &
Suc (Ant b3) = 'not' b
5 &
'not' (Suc b3) = Suc b
4 & b
3 = (a1 . b2) `1 & b
4 = (a1 . b1) `1 &
(Ant (Ant b3)) ^ <*b5*> = (a1 . a2) `1 )
if (a1 . a2) `2 = 4
ex b
1, b
2 being
Natex b
3, b
4 being
FinSequence of
CQC-WFF st
( 1
<= b
1 & b
1 < a
2 & 1
<= b
2 & b
2 < b
1 &
Ant b
3 = Ant b
4 & b
3 = (a1 . b2) `1 & b
4 = (a1 . b1) `1 &
(Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (a1 . a2) `1 )
if (a1 . a2) `2 = 5
ex b
1 being
Natex b
2 being
FinSequence of
CQC-WFF ex b
3, b
4 being
Element of
CQC-WFF st
( 1
<= b
1 & b
1 < a
2 & b
3 '&' b
4 = Suc b
2 & b
2 = (a1 . b1) `1 &
(Ant b2) ^ <*b3*> = (a1 . a2) `1 )
if (a1 . a2) `2 = 6
ex b
1 being
Natex b
2 being
FinSequence of
CQC-WFF ex b
3, b
4 being
Element of
CQC-WFF st
( 1
<= b
1 & b
1 < a
2 & b
3 '&' b
4 = Suc b
2 & b
2 = (a1 . b1) `1 &
(Ant b2) ^ <*b4*> = (a1 . a2) `1 )
if (a1 . a2) `2 = 7
ex b
1 being
Natex b
2 being
FinSequence of
CQC-WFF ex b
3 being
Element of
CQC-WFF ex b
4, b
5 being
bound_QC-variable st
( 1
<= b
1 & b
1 < a
2 &
Suc b
2 = All b
4,b
3 & b
2 = (a1 . b1) `1 &
(Ant b2) ^ <*(b3 . b4,b5)*> = (a1 . a2) `1 )
if (a1 . a2) `2 = 8
ex b
1 being
Natex b
2 being
FinSequence of
CQC-WFF ex b
3 being
Element of
CQC-WFF ex b
4, b
5 being
bound_QC-variable st
( 1
<= b
1 & b
1 < a
2 &
Suc b
2 = b
3 . b
4,b
5 & not b
5 in still_not-bound_in (Ant b2) & not b
5 in still_not-bound_in (All b4,b3) & b
2 = (a1 . b1) `1 &
(Ant b2) ^ <*(All b4,b3)*> = (a1 . a2) `1 )
if (a1 . a2) `2 = 9
;
consistency
( ( (c1 . c2) `2 = 0 & (c1 . c2) `2 = 1 implies ( not ( ex b1 being FinSequence of CQC-WFF st
( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) & ( for b1 being FinSequence of CQC-WFF holds
not (c1 . c2) `1 = b1 ^ <*VERUM *> ) ) & not ( ex b1 being FinSequence of CQC-WFF st (c1 . c2) `1 = b1 ^ <*VERUM *> & ( for b1 being FinSequence of CQC-WFF holds
not ( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) ) ) ) ) & ( (c1 . c2) `2 = 0 & (c1 . c2) `2 = 2 implies ( not ( ex b1 being FinSequence of CQC-WFF st
( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) & ( for b1 being Nat
for b2, b3 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) ) ) & not ( ex b1 being Natex b2, b3 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) & ( for b1 being FinSequence of CQC-WFF holds
not ( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) ) ) ) ) & ( (c1 . c2) `2 = 0 & (c1 . c2) `2 = 3 implies ( not ( ex b1 being FinSequence of CQC-WFF st
( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not ( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) ) ) ) ) & ( (c1 . c2) `2 = 0 & (c1 . c2) `2 = 4 implies ( not ( ex b1 being FinSequence of CQC-WFF st
( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF
for b5 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF ex b5 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not ( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) ) ) ) ) & ( (c1 . c2) `2 = 0 & (c1 . c2) `2 = 5 implies ( not ( ex b1 being FinSequence of CQC-WFF st
( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not ( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) ) ) ) ) & ( (c1 . c2) `2 = 0 & (c1 . c2) `2 = 6 implies ( not ( ex b1 being FinSequence of CQC-WFF st
( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not ( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) ) ) ) ) & ( (c1 . c2) `2 = 0 & (c1 . c2) `2 = 7 implies ( not ( ex b1 being FinSequence of CQC-WFF st
( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not ( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) ) ) ) ) & ( (c1 . c2) `2 = 0 & (c1 . c2) `2 = 8 implies ( not ( ex b1 being FinSequence of CQC-WFF st
( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not ( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) ) ) ) ) & ( (c1 . c2) `2 = 0 & (c1 . c2) `2 = 9 implies ( not ( ex b1 being FinSequence of CQC-WFF st
( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not ( Suc b1 is_tail_of Ant b1 & (c1 . c2) `1 = b1 ) ) ) ) ) & ( (c1 . c2) `2 = 1 & (c1 . c2) `2 = 2 implies ( not ( ex b1 being FinSequence of CQC-WFF st (c1 . c2) `1 = b1 ^ <*VERUM *> & ( for b1 being Nat
for b2, b3 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) ) ) & not ( ex b1 being Natex b2, b3 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) & ( for b1 being FinSequence of CQC-WFF holds
not (c1 . c2) `1 = b1 ^ <*VERUM *> ) ) ) ) & ( (c1 . c2) `2 = 1 & (c1 . c2) `2 = 3 implies ( not ( ex b1 being FinSequence of CQC-WFF st (c1 . c2) `1 = b1 ^ <*VERUM *> & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not (c1 . c2) `1 = b1 ^ <*VERUM *> ) ) ) ) & ( (c1 . c2) `2 = 1 & (c1 . c2) `2 = 4 implies ( not ( ex b1 being FinSequence of CQC-WFF st (c1 . c2) `1 = b1 ^ <*VERUM *> & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF
for b5 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF ex b5 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not (c1 . c2) `1 = b1 ^ <*VERUM *> ) ) ) ) & ( (c1 . c2) `2 = 1 & (c1 . c2) `2 = 5 implies ( not ( ex b1 being FinSequence of CQC-WFF st (c1 . c2) `1 = b1 ^ <*VERUM *> & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not (c1 . c2) `1 = b1 ^ <*VERUM *> ) ) ) ) & ( (c1 . c2) `2 = 1 & (c1 . c2) `2 = 6 implies ( not ( ex b1 being FinSequence of CQC-WFF st (c1 . c2) `1 = b1 ^ <*VERUM *> & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not (c1 . c2) `1 = b1 ^ <*VERUM *> ) ) ) ) & ( (c1 . c2) `2 = 1 & (c1 . c2) `2 = 7 implies ( not ( ex b1 being FinSequence of CQC-WFF st (c1 . c2) `1 = b1 ^ <*VERUM *> & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not (c1 . c2) `1 = b1 ^ <*VERUM *> ) ) ) ) & ( (c1 . c2) `2 = 1 & (c1 . c2) `2 = 8 implies ( not ( ex b1 being FinSequence of CQC-WFF st (c1 . c2) `1 = b1 ^ <*VERUM *> & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not (c1 . c2) `1 = b1 ^ <*VERUM *> ) ) ) ) & ( (c1 . c2) `2 = 1 & (c1 . c2) `2 = 9 implies ( not ( ex b1 being FinSequence of CQC-WFF st (c1 . c2) `1 = b1 ^ <*VERUM *> & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) & ( for b1 being FinSequence of CQC-WFF holds
not (c1 . c2) `1 = b1 ^ <*VERUM *> ) ) ) ) & ( (c1 . c2) `2 = 2 & (c1 . c2) `2 = 3 implies ( not ( ex b1 being Natex b2, b3 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2, b3 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) ) ) ) ) & ( (c1 . c2) `2 = 2 & (c1 . c2) `2 = 4 implies ( not ( ex b1 being Natex b2, b3 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF
for b5 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF ex b5 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2, b3 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) ) ) ) ) & ( (c1 . c2) `2 = 2 & (c1 . c2) `2 = 5 implies ( not ( ex b1 being Natex b2, b3 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2, b3 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) ) ) ) ) & ( (c1 . c2) `2 = 2 & (c1 . c2) `2 = 6 implies ( not ( ex b1 being Natex b2, b3 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2, b3 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) ) ) ) ) & ( (c1 . c2) `2 = 2 & (c1 . c2) `2 = 7 implies ( not ( ex b1 being Natex b2, b3 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2, b3 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) ) ) ) ) & ( (c1 . c2) `2 = 2 & (c1 . c2) `2 = 8 implies ( not ( ex b1 being Natex b2, b3 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2, b3 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) ) ) ) ) & ( (c1 . c2) `2 = 2 & (c1 . c2) `2 = 9 implies ( not ( ex b1 being Natex b2, b3 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2, b3 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & Ant b2 is_Subsequence_of Ant b3 & Suc b2 = Suc b3 & (c1 . b1) `1 = b2 & (c1 . c2) `1 = b3 ) ) ) ) ) & ( (c1 . c2) `2 = 3 & (c1 . c2) `2 = 4 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF
for b5 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF ex b5 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 3 & (c1 . c2) `2 = 5 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 3 & (c1 . c2) `2 = 6 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 3 & (c1 . c2) `2 = 7 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 3 & (c1 . c2) `2 = 8 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 3 & (c1 . c2) `2 = 9 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & len b4 > 1 & Ant (Ant b3) = Ant (Ant b4) & 'not' (Suc (Ant b3)) = Suc (Ant b4) & Suc b3 = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*(Suc b3)*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 4 & (c1 . c2) `2 = 5 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF ex b5 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) ) ) & not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF
for b5 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 4 & (c1 . c2) `2 = 6 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF ex b5 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF
for b5 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 4 & (c1 . c2) `2 = 7 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF ex b5 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF
for b5 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 4 & (c1 . c2) `2 = 8 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF ex b5 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF
for b5 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 4 & (c1 . c2) `2 = 9 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF ex b5 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF
for b5 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & len b3 > 1 & Ant b3 = Ant b4 & Suc (Ant b3) = 'not' b5 & 'not' (Suc b3) = Suc b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant (Ant b3)) ^ <*b5*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 5 & (c1 . c2) `2 = 6 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 5 & (c1 . c2) `2 = 7 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 5 & (c1 . c2) `2 = 8 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 5 & (c1 . c2) `2 = 9 implies ( not ( ex b1, b2 being Natex b3, b4 being FinSequence of CQC-WFF st
( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) & ( for b1, b2 being Nat
for b3, b4 being FinSequence of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & 1 <= b2 & b2 < b1 & Ant b3 = Ant b4 & b3 = (c1 . b2) `1 & b4 = (c1 . b1) `1 & (Ant b3) ^ <*((Suc b3) '&' (Suc b4))*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 6 & (c1 . c2) `2 = 7 implies ( not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 6 & (c1 . c2) `2 = 8 implies ( not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 6 & (c1 . c2) `2 = 9 implies ( not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b3*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 7 & (c1 . c2) `2 = 8 implies ( not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 7 & (c1 . c2) `2 = 9 implies ( not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3, b4 being Element of CQC-WFF st
( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3, b4 being Element of CQC-WFF holds
not ( 1 <= b1 & b1 < c2 & b3 '&' b4 = Suc b2 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*b4*> = (c1 . c2) `1 ) ) ) ) ) & ( (c1 . c2) `2 = 8 & (c1 . c2) `2 = 9 implies ( not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) ) ) & not ( ex b1 being Natex b2 being FinSequence of CQC-WFF ex b3 being Element of CQC-WFF ex b4, b5 being bound_QC-variable st
( 1 <= b1 & b1 < c2 & Suc b2 = b3 . b4,b5 & not b5 in still_not-bound_in (Ant b2) & not b5 in still_not-bound_in (All b4,b3) & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(All b4,b3)*> = (c1 . c2) `1 ) & ( for b1 being Nat
for b2 being FinSequence of CQC-WFF
for b3 being Element of CQC-WFF
for b4, b5 being bound_QC-variable holds
not ( 1 <= b1 & b1 < c2 & Suc b2 = All b4,b3 & b2 = (c1 . b1) `1 & (Ant b2) ^ <*(b3 . b4,b5)*> = (c1 . c2) `1 ) ) ) ) ) )
;
end;
:: deftheorem Def7 defines is_a_correct_step CALCUL_1:def 7 :
for b
1 being
FinSequence of
[:set_of_CQC-WFF-seq ,Proof_Step_Kinds :]for b
2 being
Nat holds
( (
(b1 . b2) `2 = 0 implies ( b
1,b
2 is_a_correct_step iff ex b
3 being
FinSequence of
CQC-WFF st
(
Suc b
3 is_tail_of Ant b
3 &
(b1 . b2) `1 = b
3 ) ) ) & (
(b1 . b2) `2 = 1 implies ( b
1,b
2 is_a_correct_step iff ex b
3 being
FinSequence of
CQC-WFF st
(b1 . b2) `1 = b
3 ^ <*VERUM *> ) ) & (
(b1 . b2) `2 = 2 implies ( b
1,b
2 is_a_correct_step iff ex b
3 being
Natex b
4, b
5 being
FinSequence of
CQC-WFF st
( 1
<= b
3 & b
3 < b
2 &
Ant b
4 is_Subsequence_of Ant b
5 &
Suc b
4 = Suc b
5 &
(b1 . b3) `1 = b
4 &
(b1 . b2) `1 = b
5 ) ) ) & (
(b1 . b2) `2 = 3 implies ( b
1,b
2 is_a_correct_step iff ex b
3, b
4 being
Natex b
5, b
6 being
FinSequence of
CQC-WFF st
( 1
<= b
3 & b
3 < b
2 & 1
<= b
4 & b
4 < b
3 &
len b
5 > 1 &
len b
6 > 1 &
Ant (Ant b5) = Ant (Ant b6) &
'not' (Suc (Ant b5)) = Suc (Ant b6) &
Suc b
5 = Suc b
6 & b
5 = (b1 . b4) `1 & b
6 = (b1 . b3) `1 &
(Ant (Ant b5)) ^ <*(Suc b5)*> = (b1 . b2) `1 ) ) ) & (
(b1 . b2) `2 = 4 implies ( b
1,b
2 is_a_correct_step iff ex b
3, b
4 being
Natex b
5, b
6 being
FinSequence of
CQC-WFF ex b
7 being
Element of
CQC-WFF st
( 1
<= b
3 & b
3 < b
2 & 1
<= b
4 & b
4 < b
3 &
len b
5 > 1 &
Ant b
5 = Ant b
6 &
Suc (Ant b5) = 'not' b
7 &
'not' (Suc b5) = Suc b
6 & b
5 = (b1 . b4) `1 & b
6 = (b1 . b3) `1 &
(Ant (Ant b5)) ^ <*b7*> = (b1 . b2) `1 ) ) ) & (
(b1 . b2) `2 = 5 implies ( b
1,b
2 is_a_correct_step iff ex b
3, b
4 being
Natex b
5, b
6 being
FinSequence of
CQC-WFF st
( 1
<= b
3 & b
3 < b
2 & 1
<= b
4 & b
4 < b
3 &
Ant b
5 = Ant b
6 & b
5 = (b1 . b4) `1 & b
6 = (b1 . b3) `1 &
(Ant b5) ^ <*((Suc b5) '&' (Suc b6))*> = (b1 . b2) `1 ) ) ) & (
(b1 . b2) `2 = 6 implies ( b
1,b
2 is_a_correct_step iff ex b
3 being
Natex b
4 being
FinSequence of
CQC-WFF ex b
5, b
6 being
Element of
CQC-WFF st
( 1
<= b
3 & b
3 < b
2 & b
5 '&' b
6 = Suc b
4 & b
4 = (b1 . b3) `1 &
(Ant b4) ^ <*b5*> = (b1 . b2) `1 ) ) ) & (
(b1 . b2) `2 = 7 implies ( b
1,b
2 is_a_correct_step iff ex b
3 being
Natex b
4 being
FinSequence of
CQC-WFF ex b
5, b
6 being
Element of
CQC-WFF st
( 1
<= b
3 & b
3 < b
2 & b
5 '&' b
6 = Suc b
4 & b
4 = (b1 . b3) `1 &
(Ant b4) ^ <*b6*> = (b1 . b2) `1 ) ) ) & (
(b1 . b2) `2 = 8 implies ( b
1,b
2 is_a_correct_step iff ex b
3 being
Natex b
4 being
FinSequence of
CQC-WFF ex b
5 being
Element of
CQC-WFF ex b
6, b
7 being
bound_QC-variable st
( 1
<= b
3 & b
3 < b
2 &
Suc b
4 = All b
6,b
5 & b
4 = (b1 . b3) `1 &
(Ant b4) ^ <*(b5 . b6,b7)*> = (b1 . b2) `1 ) ) ) & (
(b1 . b2) `2 = 9 implies ( b
1,b
2 is_a_correct_step iff ex b
3 being
Natex b
4 being
FinSequence of
CQC-WFF ex b
5 being
Element of
CQC-WFF ex b
6, b
7 being
bound_QC-variable st
( 1
<= b
3 & b
3 < b
2 &
Suc b
4 = b
5 . b
6,b
7 & not b
7 in still_not-bound_in (Ant b4) & not b
7 in still_not-bound_in (All b6,b5) & b
4 = (b1 . b3) `1 &
(Ant b4) ^ <*(All b6,b5)*> = (b1 . b2) `1 ) ) ) );
:: deftheorem Def8 defines a_proof CALCUL_1:def 8 :
:: deftheorem Def9 defines |- CALCUL_1:def 9 :
:: deftheorem Def10 defines is_formal_provable_from CALCUL_1:def 10 :
:: deftheorem Def11 defines |= CALCUL_1:def 11 :
:: deftheorem Def12 defines |= CALCUL_1:def 12 :
:: deftheorem Def13 defines |= CALCUL_1:def 13 :
:: deftheorem Def14 defines |= CALCUL_1:def 14 :
:: deftheorem Def15 defines |= CALCUL_1:def 15 :
theorem Th15: :: CALCUL_1:15
theorem Th16: :: CALCUL_1:16
theorem Th17: :: CALCUL_1:17
theorem Th18: :: CALCUL_1:18
theorem Th19: :: CALCUL_1:19
theorem Th20: :: CALCUL_1:20
theorem Th21: :: CALCUL_1:21
theorem Th22: :: CALCUL_1:22
theorem Th23: :: CALCUL_1:23
theorem Th24: :: CALCUL_1:24
theorem Th25: :: CALCUL_1:25
theorem Th26: :: CALCUL_1:26
theorem Th27: :: CALCUL_1:27
theorem Th28: :: CALCUL_1:28
theorem Th29: :: CALCUL_1:29
theorem Th30: :: CALCUL_1:30
theorem Th31: :: CALCUL_1:31
Lemma45:
for b1 being FinSequence holds
( b1 <> {} implies 1 <= len b1 )
by GRAPH_5:8;
Lemma46:
for b1 being Nat
for b2 being FinSequence of [:set_of_CQC-WFF-seq ,Proof_Step_Kinds :] holds
( 1 <= b1 & b1 <= len b2 implies (b2 . b1) `1 is FinSequence of CQC-WFF )
theorem Th32: :: CALCUL_1:32
theorem Th33: :: CALCUL_1:33
theorem Th34: :: CALCUL_1:34
theorem Th35: :: CALCUL_1:35
theorem Th36: :: CALCUL_1:36
theorem Th37: :: CALCUL_1:37
theorem Th38: :: CALCUL_1:38
theorem Th39: :: CALCUL_1:39
theorem Th40: :: CALCUL_1:40
theorem Th41: :: CALCUL_1:41
theorem Th42: :: CALCUL_1:42
theorem Th43: :: CALCUL_1:43
theorem Th44: :: CALCUL_1:44
theorem Th45: :: CALCUL_1:45
theorem Th46: :: CALCUL_1:46
theorem Th47: :: CALCUL_1:47
theorem Th48: :: CALCUL_1:48
theorem Th49: :: CALCUL_1:49
theorem Th50: :: CALCUL_1:50
theorem Th51: :: CALCUL_1:51
theorem Th52: :: CALCUL_1:52
theorem Th53: :: CALCUL_1:53
theorem Th54: :: CALCUL_1:54
theorem Th55: :: CALCUL_1:55
theorem Th56: :: CALCUL_1:56
theorem Th57: :: CALCUL_1:57
theorem Th58: :: CALCUL_1:58
theorem Th59: :: CALCUL_1:59
theorem Th60: :: CALCUL_1:60
theorem Th61: :: CALCUL_1:61
theorem Th62: :: CALCUL_1:62
theorem Th63: :: CALCUL_1:63
theorem Th64: :: CALCUL_1:64
theorem Th65: :: CALCUL_1:65
theorem Th66: :: CALCUL_1:66
theorem Th67: :: CALCUL_1:67
theorem Th68: :: CALCUL_1:68