QC-WFF is Subset of ([:NAT,NAT:] *) by QC_LANG1:5, QC_LANG1:def 8;
hence [:QC-WFF,vSUB:] is Subset of [:([:NAT,NAT:] *),vSUB:] by ZFMISC_1:95; :: thesis: ( ( for k being Element of NAT
for p being QC-pred_symbol of k
for ll being QC-variable_list of k
for e being Element of vSUB holds [(<*p*> ^ ll),e] in [:QC-WFF,vSUB:] ) & ( for e being Element of vSUB holds [<*[0,0]*>,e] in [:QC-WFF,vSUB:] ) & ( for p being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,e] in [:QC-WFF,vSUB:] holds
[(<*[1,0]*> ^ p),e] in [:QC-WFF,vSUB:] ) & ( for p, q being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,e] in [:QC-WFF,vSUB:] & [q,e] in [:QC-WFF,vSUB:] holds
[((<*[2,0]*> ^ p) ^ q),e] in [:QC-WFF,vSUB:] ) & ( for x being bound_QC-variable
for p being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,(QSub . [((<*[3,0]*> ^ <*x*>) ^ p),e])] in [:QC-WFF,vSUB:] holds
[((<*[3,0]*> ^ <*x*>) ^ p),e] in [:QC-WFF,vSUB:] ) )
thus for k being Element of NAT
for p being QC-pred_symbol of k
for ll being QC-variable_list of k
for e being Element of vSUB holds [(<*p*> ^ ll),e] in [:QC-WFF,vSUB:] :: thesis: ( ( for e being Element of vSUB holds [<*[0,0]*>,e] in [:QC-WFF,vSUB:] ) & ( for p being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,e] in [:QC-WFF,vSUB:] holds
[(<*[1,0]*> ^ p),e] in [:QC-WFF,vSUB:] ) & ( for p, q being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,e] in [:QC-WFF,vSUB:] & [q,e] in [:QC-WFF,vSUB:] holds
[((<*[2,0]*> ^ p) ^ q),e] in [:QC-WFF,vSUB:] ) & ( for x being bound_QC-variable
for p being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,(QSub . [((<*[3,0]*> ^ <*x*>) ^ p),e])] in [:QC-WFF,vSUB:] holds
[((<*[3,0]*> ^ <*x*>) ^ p),e] in [:QC-WFF,vSUB:] ) )

proof

let k be Element of NAT ; :: thesis: for p being QC-pred_symbol of k
for ll being QC-variable_list of k
for e being Element of vSUB holds [(<*p*> ^ ll),e] in [:QC-WFF,vSUB:]
let p be QC-pred_symbol of k; :: thesis: for ll being QC-variable_list of k
for e being Element of vSUB holds [(<*p*> ^ ll),e] in [:QC-WFF,vSUB:]
let ll be QC-variable_list of k; :: thesis: for e being Element of vSUB holds [(<*p*> ^ ll),e] in [:QC-WFF,vSUB:]
let e be Element of vSUB ; :: thesis: [(<*p*> ^ ll),e] in [:QC-WFF,vSUB:]
p ! ll = <*p*> ^ ll by QC_LANG1:6;
hence [(<*p*> ^ ll),e] in [:QC-WFF,vSUB:] by ZFMISC_1:def 2; :: thesis: verum

end;

thus for e being Element of vSUB holds [<*[0,0]*>,e] in [:QC-WFF,vSUB:] by QC_LANG1:def 12, ZFMISC_1:def 2; :: thesis: ( ( for p being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,e] in [:QC-WFF,vSUB:] holds
[(<*[1,0]*> ^ p),e] in [:QC-WFF,vSUB:] ) & ( for p, q being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,e] in [:QC-WFF,vSUB:] & [q,e] in [:QC-WFF,vSUB:] holds
[((<*[2,0]*> ^ p) ^ q),e] in [:QC-WFF,vSUB:] ) & ( for x being bound_QC-variable
for p being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,(QSub . [((<*[3,0]*> ^ <*x*>) ^ p),e])] in [:QC-WFF,vSUB:] holds
[((<*[3,0]*> ^ <*x*>) ^ p),e] in [:QC-WFF,vSUB:] ) )
thus for p being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,e] in [:QC-WFF,vSUB:] holds
[(<*[1,0]*> ^ p),e] in [:QC-WFF,vSUB:] :: thesis: ( ( for p, q being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,e] in [:QC-WFF,vSUB:] & [q,e] in [:QC-WFF,vSUB:] holds
[((<*[2,0]*> ^ p) ^ q),e] in [:QC-WFF,vSUB:] ) & ( for x being bound_QC-variable
for p being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,(QSub . [((<*[3,0]*> ^ <*x*>) ^ p),e])] in [:QC-WFF,vSUB:] holds
[((<*[3,0]*> ^ <*x*>) ^ p),e] in [:QC-WFF,vSUB:] ) )

proof

let p be FinSequence of [:NAT,NAT:]; :: thesis: for e being Element of vSUB st [p,e] in [:QC-WFF,vSUB:] holds
[(<*[1,0]*> ^ p),e] in [:QC-WFF,vSUB:]
let e be Element of vSUB ; :: thesis: ( [p,e] in [:QC-WFF,vSUB:] implies [(<*[1,0]*> ^ p),e] in [:QC-WFF,vSUB:] )
assume [p,e] in [:QC-WFF,vSUB:] ; :: thesis: [(<*[1,0]*> ^ p),e] in [:QC-WFF,vSUB:]
then ex a, b being set st
( a in QC-WFF & b in vSUB & [p,e] = [a,b] ) by ZFMISC_1:def 2;
then reconsider p9 = p as Element of QC-WFF by ZFMISC_1:27;
'not' p9 = <*[1,0]*> ^ (@ p9) ;
hence [(<*[1,0]*> ^ p),e] in [:QC-WFF,vSUB:] by ZFMISC_1:def 2; :: thesis: verum

end;

thus for p, q being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,e] in [:QC-WFF,vSUB:] & [q,e] in [:QC-WFF,vSUB:] holds
[((<*[2,0]*> ^ p) ^ q),e] in [:QC-WFF,vSUB:] :: thesis: for x being bound_QC-variable
for p being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,(QSub . [((<*[3,0]*> ^ <*x*>) ^ p),e])] in [:QC-WFF,vSUB:] holds
[((<*[3,0]*> ^ <*x*>) ^ p),e] in [:QC-WFF,vSUB:]

proof

let p, q be FinSequence of [:NAT,NAT:]; :: thesis: for e being Element of vSUB st [p,e] in [:QC-WFF,vSUB:] & [q,e] in [:QC-WFF,vSUB:] holds
[((<*[2,0]*> ^ p) ^ q),e] in [:QC-WFF,vSUB:]
let e be Element of vSUB ; :: thesis: ( [p,e] in [:QC-WFF,vSUB:] & [q,e] in [:QC-WFF,vSUB:] implies [((<*[2,0]*> ^ p) ^ q),e] in [:QC-WFF,vSUB:] )
assume that
A1: [p,e] in [:QC-WFF,vSUB:] and
A2: [q,e] in [:QC-WFF,vSUB:] ; :: thesis: [((<*[2,0]*> ^ p) ^ q),e] in [:QC-WFF,vSUB:]
ex c, d being set st
( c in QC-WFF & d in vSUB & [q,e] = [c,d] ) by A2, ZFMISC_1:def 2;
then reconsider q9 = q as Element of QC-WFF by ZFMISC_1:27;
ex a, b being set st
( a in QC-WFF & b in vSUB & [p,e] = [a,b] ) by A1, ZFMISC_1:def 2;
then reconsider p9 = p as Element of QC-WFF by ZFMISC_1:27;
p9 '&' q9 = (<*[2,0]*> ^ (@ p9)) ^ (@ q9) ;
hence [((<*[2,0]*> ^ p) ^ q),e] in [:QC-WFF,vSUB:] by ZFMISC_1:def 2; :: thesis: verum

end;

thus for x being bound_QC-variable
for p being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,(QSub . [((<*[3,0]*> ^ <*x*>) ^ p),e])] in [:QC-WFF,vSUB:] holds
[((<*[3,0]*> ^ <*x*>) ^ p),e] in [:QC-WFF,vSUB:] :: thesis: verum

proof

let x be bound_QC-variable; :: thesis: for p being FinSequence of [:NAT,NAT:]
for e being Element of vSUB st [p,(QSub . [((<*[3,0]*> ^ <*x*>) ^ p),e])] in [:QC-WFF,vSUB:] holds
[((<*[3,0]*> ^ <*x*>) ^ p),e] in [:QC-WFF,vSUB:]
let p be FinSequence of [:NAT,NAT:]; :: thesis: for e being Element of vSUB st [p,(QSub . [((<*[3,0]*> ^ <*x*>) ^ p),e])] in [:QC-WFF,vSUB:] holds
[((<*[3,0]*> ^ <*x*>) ^ p),e] in [:QC-WFF,vSUB:]
let e be Element of vSUB ; :: thesis: ( [p,(QSub . [((<*[3,0]*> ^ <*x*>) ^ p),e])] in [:QC-WFF,vSUB:] implies [((<*[3,0]*> ^ <*x*>) ^ p),e] in [:QC-WFF,vSUB:] )
assume [p,(QSub . [((<*[3,0]*> ^ <*x*>) ^ p),e])] in [:QC-WFF,vSUB:] ; :: thesis: [((<*[3,0]*> ^ <*x*>) ^ p),e] in [:QC-WFF,vSUB:]
then ex a, b being set st
( a in QC-WFF & b in vSUB & [p,(QSub . [((<*[3,0]*> ^ <*x*>) ^ p),e])] = [a,b] ) by ZFMISC_1:def 2;
then reconsider p9 = p as Element of QC-WFF by ZFMISC_1:27;
All (x,p9) = (<*[3,0]*> ^ <*x*>) ^ (@ p9) ;
hence [((<*[3,0]*> ^ <*x*>) ^ p),e] in [:QC-WFF,vSUB:] by ZFMISC_1:def 2; :: thesis: verum

end;