let Al be QC-alphabet ; :: thesis: for X being Subset of (CQC-WFF Al)
for p being Element of CQC-WFF Al st p in Cn X holds
ex Y being Subset of (CQC-WFF Al) st
( Y c= X & Y is finite & p in Cn Y )

let X be Subset of (CQC-WFF Al); :: thesis: for p being Element of CQC-WFF Al st p in Cn X holds
ex Y being Subset of (CQC-WFF Al) st
( Y c= X & Y is finite & p in Cn Y )

let p be Element of CQC-WFF Al; :: thesis: ( p in Cn X implies ex Y being Subset of (CQC-WFF Al) st
( Y c= X & Y is finite & p in Cn Y ) )

assume p in Cn X ; :: thesis: ex Y being Subset of (CQC-WFF Al) st
( Y c= X & Y is finite & p in Cn Y )

then consider f being FinSequence of such that
A1: f is_a_proof_wrt X and
A2: Effect f = p by Th32;
A3: f <> {} by A1;
consider A being set such that
A4: A is finite and
A5: A c= CQC-WFF Al and
A6: rng f c= by ;
reconsider Z = A as Subset of (CQC-WFF Al) by A5;
take Y = Z /\ X; :: thesis: ( Y c= X & Y is finite & p in Cn Y )
thus Y c= X by XBOOLE_1:17; :: thesis: ( Y is finite & p in Cn Y )
thus Y is finite by A4; :: thesis: p in Cn Y
for n being Nat st 1 <= n & n <= len f holds
f,n is_a_correct_step_wrt Y
proof
let n be Nat; :: thesis: ( 1 <= n & n <= len f implies f,n is_a_correct_step_wrt Y )
assume A7: ( 1 <= n & n <= len f ) ; :: thesis:
then A8: f,n is_a_correct_step_wrt X by A1;
not not (f . n) `2 = 0 & ... & not (f . n) `2 = 9 by ;
per cases ) `2 = 0 or (f . n) `2 = 1 or (f . n) `2 = 2 or (f . n) `2 = 3 or (f . n) `2 = 4 or (f . n) `2 = 5 or (f . n) `2 = 6 or (f . n) `2 = 7 or (f . n) `2 = 8 or (f . n) `2 = 9 ) ;
:: according to CQC_THE1:def 4
case (f . n) `2 = 0 ; :: thesis: (f . n) `1 in Y
then A9: (f . n) `1 in X by ;
n in Seg (len f) by ;
then n in dom f by FINSEQ_1:def 3;
then f . n in rng f by FUNCT_1:def 3;
then (f . n) `1 in A by ;
hence (f . n) `1 in Y by ; :: thesis: verum
end;
case (f . n) `2 = 1 ; :: thesis: (f . n) `1 = VERUM Al
hence (f . n) `1 = VERUM Al by ; :: thesis: verum
end;
case (f . n) `2 = 2 ; :: thesis: ex p being Element of CQC-WFF Al st (f . n) `1 = (() => p) => p
hence ex p being Element of CQC-WFF Al st (f . n) `1 = (() => p) => p by ; :: thesis: verum
end;
case (f . n) `2 = 3 ; :: thesis: ex p, q being Element of CQC-WFF Al st (f . n) `1 = p => (() => q)
hence ex p, q being Element of CQC-WFF Al st (f . n) `1 = p => (() => q) by ; :: thesis: verum
end;
case (f . n) `2 = 4 ; :: thesis: ex p, q, r being Element of CQC-WFF Al st (f . n) `1 = (p => q) => (('not' (q '&' r)) => ('not' (p '&' r)))
hence ex p, q, r being Element of CQC-WFF Al st (f . n) `1 = (p => q) => (('not' (q '&' r)) => ('not' (p '&' r))) by ; :: thesis: verum
end;
case (f . n) `2 = 5 ; :: thesis: ex p, q being Element of CQC-WFF Al st (f . n) `1 = (p '&' q) => (q '&' p)
hence ex p, q being Element of CQC-WFF Al st (f . n) `1 = (p '&' q) => (q '&' p) by ; :: thesis: verum
end;
case (f . n) `2 = 6 ; :: thesis: ex p being Element of CQC-WFF Al ex x being bound_QC-variable of Al st (f . n) `1 = (All (x,p)) => p
hence ex p being Element of CQC-WFF Al ex x being bound_QC-variable of Al st (f . n) `1 = (All (x,p)) => p by ; :: thesis: verum
end;
case (f . n) `2 = 7 ; :: thesis: ex i, j being Nat ex p, q being Element of CQC-WFF Al st
( 1 <= i & i < n & 1 <= j & j < i & p = (f . j) `1 & q = (f . n) `1 & (f . i) `1 = p => q )

hence ex i, j being Nat ex p, q being Element of CQC-WFF Al st
( 1 <= i & i < n & 1 <= j & j < i & p = (f . j) `1 & q = (f . n) `1 & (f . i) `1 = p => q ) by ; :: thesis: verum
end;
case (f . n) `2 = 8 ; :: thesis: ex i being Nat ex p, q being Element of CQC-WFF Al ex x being bound_QC-variable of Al st
( 1 <= i & i < n & (f . i) `1 = p => q & not x in still_not-bound_in p & (f . n) `1 = p => (All (x,q)) )

hence ex i being Nat ex p, q being Element of CQC-WFF Al ex x being bound_QC-variable of Al st
( 1 <= i & i < n & (f . i) `1 = p => q & not x in still_not-bound_in p & (f . n) `1 = p => (All (x,q)) ) by ; :: thesis: verum
end;
case (f . n) `2 = 9 ; :: thesis: ex i being Nat ex x, y being bound_QC-variable of Al ex s being QC-formula of Al st
( 1 <= i & i < n & s . x in CQC-WFF Al & s . y in CQC-WFF Al & not x in still_not-bound_in s & s . x = (f . i) `1 & s . y = (f . n) `1 )

hence ex i being Nat ex x, y being bound_QC-variable of Al ex s being QC-formula of Al st
( 1 <= i & i < n & s . x in CQC-WFF Al & s . y in CQC-WFF Al & not x in still_not-bound_in s & s . x = (f . i) `1 & s . y = (f . n) `1 ) by ; :: thesis: verum
end;
end;
end;
then f is_a_proof_wrt Y by A3;
then p in { q where q is Element of CQC-WFF Al : ex f being FinSequence of st
( f is_a_proof_wrt Y & Effect f = q )
}
by A2;
hence p in Cn Y by Th31; :: thesis: verum