let Al be QC-alphabet ; :: thesis: for X, Y being Subset of (CQC-WFF Al)
for f being FinSequence of st f is_a_proof_wrt X & X c= Y holds
f is_a_proof_wrt Y

let X, Y be Subset of (CQC-WFF Al); :: thesis: for f being FinSequence of st f is_a_proof_wrt X & X c= Y holds
f is_a_proof_wrt Y

let f be FinSequence of ; :: thesis: ( f is_a_proof_wrt X & X c= Y implies f is_a_proof_wrt Y )
assume that
A1: f is_a_proof_wrt X and
A2: X c= Y ; :: thesis:
thus f <> {} by A1; :: according to CQC_THE1:def 5 :: thesis: for n being Nat st 1 <= n & n <= len f holds
f,n is_a_correct_step_wrt Y

let n be Nat; :: thesis: ( 1 <= n & n <= len f implies f,n is_a_correct_step_wrt Y )
assume A3: ( 1 <= n & n <= len f ) ; :: thesis:
then A4: 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 (f . n) `1 in X by ;
hence (f . n) `1 in Y by A2; :: 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;