let Al be QC-alphabet ; :: thesis: for X, Y being Subset of (CQC-WFF Al)

for f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] 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 [:(CQC-WFF Al),Proof_Step_Kinds:] st f is_a_proof_wrt X & X c= Y holds

f is_a_proof_wrt Y

let f be FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:]; :: 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: f is_a_proof_wrt Y

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: f,n is_a_correct_step_wrt Y

then A4: f,n is_a_correct_step_wrt X by A1;

not not (f . n) `2 = 0 & ... & not (f . n) `2 = 9 by A3, Th19;

for f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] 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 [:(CQC-WFF Al),Proof_Step_Kinds:] st f is_a_proof_wrt X & X c= Y holds

f is_a_proof_wrt Y

let f be FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:]; :: 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: f is_a_proof_wrt Y

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: f,n is_a_correct_step_wrt Y

then A4: f,n is_a_correct_step_wrt X by A1;

not not (f . n) `2 = 0 & ... & not (f . n) `2 = 9 by A3, Th19;

per cases then
( (f . n) `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 4end;

:: according to CQC_THE1:def 4

case
(f . n) `2 = 2
; :: thesis: ex p being Element of CQC-WFF Al st (f . n) `1 = (('not' p) => p) => p

hence
ex p being Element of CQC-WFF Al st (f . n) `1 = (('not' p) => p) => p
by A4, Def4; :: thesis: verum

end;case
(f . n) `2 = 3
; :: thesis: ex p, q being Element of CQC-WFF Al st (f . n) `1 = p => (('not' p) => q)

hence
ex p, q being Element of CQC-WFF Al st (f . n) `1 = p => (('not' p) => q)
by A4, Def4; :: 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 A4, Def4; :: 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 A4, Def4; :: 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 A4, Def4; :: 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 )

( 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 A4, Def4; :: thesis: verum

end;( 1 <= i & i < n & 1 <= j & j < i & p = (f . j) `1 & q = (f . n) `1 & (f . i) `1 = p => q ) by A4, Def4; :: thesis: verum

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)) )

( 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 A4, Def4; :: thesis: verum

end;( 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 A4, Def4; :: thesis: verum

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 )

( 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 A4, Def4; :: thesis: verum

end;( 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 A4, Def4; :: thesis: verum