let Al be QC-alphabet ; :: thesis: for n being Nat
for X being Subset of (CQC-WFF Al)
for f, g being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st 1 <= n & n <= len g & g,n is_a_correct_step_wrt X holds
f ^ g,n + (len f) is_a_correct_step_wrt X
let n be Nat; :: thesis: for X being Subset of (CQC-WFF Al)
for f, g being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st 1 <= n & n <= len g & g,n is_a_correct_step_wrt X holds
f ^ g,n + (len f) is_a_correct_step_wrt X
let X be Subset of (CQC-WFF Al); :: thesis: for f, g being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st 1 <= n & n <= len g & g,n is_a_correct_step_wrt X holds
f ^ g,n + (len f) is_a_correct_step_wrt X
let f, g be FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:]; :: thesis: ( 1 <= n & n <= len g & g,n is_a_correct_step_wrt X implies f ^ g,n + (len f) is_a_correct_step_wrt X )
assume that
A1: 1 <= n and
A2: n <= len g and
A3: g,n is_a_correct_step_wrt X ; :: thesis: f ^ g,n + (len f) is_a_correct_step_wrt X
n in Seg (len g) by A1, A2, FINSEQ_1:1;
then n in dom g by FINSEQ_1:def 3;
then A4: g . n = (f ^ g) . (n + (len f)) by FINSEQ_1:def 7;
n + (len f) <= (len f) + (len g) by A2, XREAL_1:6;
then n + (len f) <= len (f ^ g) by FINSEQ_1:22;
then not not ((f ^ g) . (n + (len f))) `2 = 0 & ... & not ((f ^ g) . (n + (len f))) `2 = 9 by A1, Th19, NAT_1:12;
per cases then ( ((f ^ g) . (n + (len f))) `2 = 0 or ((f ^ g) . (n + (len f))) `2 = 1 or ((f ^ g) . (n + (len f))) `2 = 2 or ((f ^ g) . (n + (len f))) `2 = 3 or ((f ^ g) . (n + (len f))) `2 = 4 or ((f ^ g) . (n + (len f))) `2 = 5 or ((f ^ g) . (n + (len f))) `2 = 6 or ((f ^ g) . (n + (len f))) `2 = 7 or ((f ^ g) . (n + (len f))) `2 = 8 or ((f ^ g) . (n + (len f))) `2 = 9 ) ;
:: according to CQC_THE1:def 4
case ((f ^ g) . (n + (len f))) `2 = 0 ; :: thesis: ((f ^ g) . (n + (len f))) `1 in X
hence ((f ^ g) . (n + (len f))) `1 in X by A3, A4, Def4; :: thesis: verum
end;
case ((f ^ g) . (n + (len f))) `2 = 1 ; :: thesis: ((f ^ g) . (n + (len f))) `1 = VERUM Al
hence ((f ^ g) . (n + (len f))) `1 = VERUM Al by A3, A4, Def4; :: thesis: verum
end;
case ((f ^ g) . (n + (len f))) `2 = 2 ; :: thesis: ex p being Element of CQC-WFF Al st ((f ^ g) . (n + (len f))) `1 = (('not' p) => p) => p
hence ex p being Element of CQC-WFF Al st ((f ^ g) . (n + (len f))) `1 = (('not' p) => p) => p by A3, A4, Def4; :: thesis: verum
end;
case ((f ^ g) . (n + (len f))) `2 = 3 ; :: thesis: ex p, q being Element of CQC-WFF Al st ((f ^ g) . (n + (len f))) `1 = p => (('not' p) => q)
hence ex p, q being Element of CQC-WFF Al st ((f ^ g) . (n + (len f))) `1 = p => (('not' p) => q) by A3, A4, Def4; :: thesis: verum
end;
case ((f ^ g) . (n + (len f))) `2 = 4 ; :: thesis: ex p, q, r being Element of CQC-WFF Al st ((f ^ g) . (n + (len f))) `1 = (p => q) => (('not' (q '&' r)) => ('not' (p '&' r)))
hence ex p, q, r being Element of CQC-WFF Al st ((f ^ g) . (n + (len f))) `1 = (p => q) => (('not' (q '&' r)) => ('not' (p '&' r))) by A3, A4, Def4; :: thesis: verum
end;
case ((f ^ g) . (n + (len f))) `2 = 5 ; :: thesis: ex p, q being Element of CQC-WFF Al st ((f ^ g) . (n + (len f))) `1 = (p '&' q) => (q '&' p)
hence ex p, q being Element of CQC-WFF Al st ((f ^ g) . (n + (len f))) `1 = (p '&' q) => (q '&' p) by A3, A4, Def4; :: thesis: verum
end;
case ((f ^ g) . (n + (len f))) `2 = 6 ; :: thesis: ex p being Element of CQC-WFF Al ex x being bound_QC-variable of Al st ((f ^ g) . (n + (len f))) `1 = (All (x,p)) => p
hence ex p being Element of CQC-WFF Al ex x being bound_QC-variable of Al st ((f ^ g) . (n + (len f))) `1 = (All (x,p)) => p by A3, A4, Def4; :: thesis: verum
end;
case ((f ^ g) . (n + (len f))) `2 = 7 ; :: thesis: ex i, j being Nat ex p, q being Element of CQC-WFF Al st
( 1 <= i & i < n + (len f) & 1 <= j & j < i & p = ((f ^ g) . j) `1 & q = ((f ^ g) . (n + (len f))) `1 & ((f ^ g) . i) `1 = p => q )
then consider i, j being Nat, r, t being Element of CQC-WFF Al such that
A5: 1 <= i and
A6: i < n and
A7: 1 <= j and
A8: j < i and
A9: ( r = (g . j) `1 & t = (g . n) `1 & (g . i) `1 = r => t ) by A3, A4, Def4;
A10: ( 1 <= i + (len f) & i + (len f) < n + (len f) ) by A5, A6, NAT_1:12, XREAL_1:6;
A11: ( 1 <= j + (len f) & j + (len f) < i + (len f) ) by A7, A8, NAT_1:12, XREAL_1:6;
A12: i <= len g by A2, A6, XXREAL_0:2;
then A13: j <= len g by A8, XXREAL_0:2;
A14: i in Seg (len g) by A5, A12, FINSEQ_1:1;
A15: j in Seg (len g) by A7, A13, FINSEQ_1:1;
A16: i in dom g by A14, FINSEQ_1:def 3;
A17: j in dom g by A15, FINSEQ_1:def 3;
A18: g . i = (f ^ g) . (i + (len f)) by A16, FINSEQ_1:def 7;
g . j = (f ^ g) . (j + (len f)) by A17, FINSEQ_1:def 7;
hence ex i, j being Nat ex p, q being Element of CQC-WFF Al st
( 1 <= i & i < n + (len f) & 1 <= j & j < i & p = ((f ^ g) . j) `1 & q = ((f ^ g) . (n + (len f))) `1 & ((f ^ g) . i) `1 = p => q ) by A4, A9, A10, A11, A18; :: thesis: verum
end;
case ((f ^ g) . (n + (len f))) `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 + (len f) & ((f ^ g) . i) `1 = p => q & not x in still_not-bound_in p & ((f ^ g) . (n + (len f))) `1 = p => (All (x,q)) )
then consider i being Nat, r, t being Element of CQC-WFF Al, x being bound_QC-variable of Al such that
A19: 1 <= i and
A20: i < n and
A21: ( (g . i) `1 = r => t & not x in still_not-bound_in r & (g . n) `1 = r => (All (x,t)) ) by A3, A4, Def4;
A22: ( 1 <= (len f) + i & (len f) + i < n + (len f) ) by A19, A20, NAT_1:12, XREAL_1:6;
i <= len g by A2, A20, XXREAL_0:2;
then i in Seg (len g) by A19, FINSEQ_1:1;
then i in dom g by FINSEQ_1:def 3;
then g . i = (f ^ g) . ((len f) + i) by FINSEQ_1:def 7;
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 + (len f) & ((f ^ g) . i) `1 = p => q & not x in still_not-bound_in p & ((f ^ g) . (n + (len f))) `1 = p => (All (x,q)) ) by A4, A21, A22; :: thesis: verum
end;
case ((f ^ g) . (n + (len f))) `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 + (len f) & s . x in CQC-WFF Al & s . y in CQC-WFF Al & not x in still_not-bound_in s & s . x = ((f ^ g) . i) `1 & s . y = ((f ^ g) . (n + (len f))) `1 )
then consider i being Nat, x, y being bound_QC-variable of Al, s being QC-formula of Al such that
A23: 1 <= i and
A24: i < n and
A25: ( s . x in CQC-WFF Al & s . y in CQC-WFF Al & not x in still_not-bound_in s & s . x = (g . i) `1 & (g . n) `1 = s . y ) by A3, A4, Def4;
A26: ( 1 <= (len f) + i & (len f) + i < n + (len f) ) by A23, A24, NAT_1:12, XREAL_1:6;
i <= len g by A2, A24, XXREAL_0:2;
then i in Seg (len g) by A23, FINSEQ_1:1;
then i in dom g by FINSEQ_1:def 3;
then g . i = (f ^ g) . ((len f) + i) by FINSEQ_1:def 7;
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 + (len f) & s . x in CQC-WFF Al & s . y in CQC-WFF Al & not x in still_not-bound_in s & s . x = ((f ^ g) . i) `1 & s . y = ((f ^ g) . (n + (len f))) `1 ) by A4, A25, A26; :: thesis: verum
end;
end;