let n be Element of NAT ; :: thesis: for X being Subset of CQC-WFF
for g, f being FinSequence of [:CQC-WFF ,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 ; :: thesis: for g, f being FinSequence of [:CQC-WFF ,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 g, f be FinSequence of [:CQC-WFF ,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 & n <= len g ) and
A2: 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, FINSEQ_1:3;
then n in dom g by FINSEQ_1:def 3;
then A3: g . n = (f ^ g) . (n + (len f)) by FINSEQ_1:def 7;
n + (len f) <= (len f) + (len g) by A1, XREAL_1:8;
then A4: ( 1 <= n + (len f) & n + (len f) <= len (f ^ g) ) by A1, FINSEQ_1:35, NAT_1:12;

per cases ( ((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 ) by A4, Th45;
:: 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 A2, A3, Def4; :: thesis: verum

end;

case ((f ^ g) . (n + (len f))) `2 = 1 ; :: thesis: ((f ^ g) . (n + (len f))) `1 = VERUM

hence ((f ^ g) . (n + (len f))) `1 = VERUM by A2, A3, Def4; :: thesis: verum

end;

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

hence ex p being Element of CQC-WFF st ((f ^ g) . (n + (len f))) `1 = (('not' p) => p) => p by A2, A3, Def4; :: thesis: verum

end;

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

hence ex p, q being Element of CQC-WFF st ((f ^ g) . (n + (len f))) `1 = p => (('not' p) => q) by A2, A3, Def4; :: thesis: verum

end;

case ((f ^ g) . (n + (len f))) `2 = 4 ; :: thesis: ex p, q, r being Element of CQC-WFF 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 st ((f ^ g) . (n + (len f))) `1 = (p => q) => (('not' (q '&' r)) => ('not' (p '&' r))) by A2, A3, Def4; :: thesis: verum

end;

case ((f ^ g) . (n + (len f))) `2 = 5 ; :: thesis: ex p, q being Element of CQC-WFF st ((f ^ g) . (n + (len f))) `1 = (p '&' q) => (q '&' p)

hence ex p, q being Element of CQC-WFF st ((f ^ g) . (n + (len f))) `1 = (p '&' q) => (q '&' p) by A2, A3, Def4; :: thesis: verum

end;

case ((f ^ g) . (n + (len f))) `2 = 6 ; :: thesis: ex p being Element of CQC-WFF ex x being bound_QC-variable st ((f ^ g) . (n + (len f))) `1 = (All x,p) => p

hence ex p being Element of CQC-WFF ex x being bound_QC-variable st ((f ^ g) . (n + (len f))) `1 = (All x,p) => p by A2, A3, Def4; :: thesis: verum

end;

case ((f ^ g) . (n + (len f))) `2 = 7 ; :: thesis: ex i, j being Element of NAT ex p, q being Element of CQC-WFF 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 Element of NAT , r, t being Element of CQC-WFF such that
A5: ( 1 <= i & i < n & 1 <= j & j < i ) and
A6: r = (g . j) `1 and
A7: t = (g . n) `1 and
A8: (g . i) `1 = r => t by A2, A3, Def4;
A9: ( 1 <= i + (len f) & i + (len f) < n + (len f) ) by A5, NAT_1:12, XREAL_1:8;
A10: ( 1 <= j + (len f) & j + (len f) < i + (len f) ) by A5, NAT_1:12, XREAL_1:8;
( i <= len g & j <= n ) by A1, A5, XXREAL_0:2;
then ( i in Seg (len g) & j <= len g ) by A5, FINSEQ_1:3, XXREAL_0:2;
then ( i in Seg (len g) & j in Seg (len g) ) by A5, FINSEQ_1:3;
then ( i in dom g & j in dom g ) by FINSEQ_1:def 3;
then ( g . i = (f ^ g) . (i + (len f)) & g . j = (f ^ g) . (j + (len f)) ) by FINSEQ_1:def 7;
hence ex i, j being Element of NAT ex p, q being Element of CQC-WFF 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 A3, A6, A7, A8, A9, A10; :: thesis: verum

end;

case ((f ^ g) . (n + (len f))) `2 = 8 ; :: thesis: ex i being Element of NAT ex p, q being Element of CQC-WFF ex x being bound_QC-variable 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 Element of NAT , r, t being Element of CQC-WFF , x being bound_QC-variable such that
A11: ( 1 <= i & i < n ) and
A12: ( (g . i) `1 = r => t & not x in still_not-bound_in r & (g . n) `1 = r => (All x,t) ) by A2, A3, Def4;
A13: ( 1 <= (len f) + i & (len f) + i < n + (len f) ) by A11, NAT_1:12, XREAL_1:8;
i <= len g by A1, A11, XXREAL_0:2;
then i in Seg (len g) by A11, FINSEQ_1:3;
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 Element of NAT ex p, q being Element of CQC-WFF ex x being bound_QC-variable 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 A3, A12, A13; :: thesis: verum

end;

case ((f ^ g) . (n + (len f))) `2 = 9 ; :: thesis: ex i being Element of NAT ex x, y being bound_QC-variable ex s being QC-formula st
( 1 <= i & i < n + (len f) & s . x in CQC-WFF & s . y in CQC-WFF & 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 Element of NAT , x, y being bound_QC-variable, s being QC-formula such that
A14: ( 1 <= i & i < n ) and
A15: ( s . x in CQC-WFF & s . y in CQC-WFF & not x in still_not-bound_in s & s . x = (g . i) `1 & (g . n) `1 = s . y ) by A2, A3, Def4;
A16: ( 1 <= (len f) + i & (len f) + i < n + (len f) ) by A14, NAT_1:12, XREAL_1:8;
i <= len g by A1, A14, XXREAL_0:2;
then i in Seg (len g) by A14, FINSEQ_1:3;
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 Element of NAT ex x, y being bound_QC-variable ex s being QC-formula st
( 1 <= i & i < n + (len f) & s . x in CQC-WFF & s . y in CQC-WFF & not x in still_not-bound_in s & s . x = ((f ^ g) . i) `1 & s . y = ((f ^ g) . (n + (len f))) `1 ) by A3, A15, A16; :: thesis: verum

end;

end;