let p, q be Element of CQC-WFF ; :: thesis: for x being bound_QC-variable
for X being Subset of CQC-WFF st p => q in { F where F is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = F ) } & not x in still_not-bound_in p holds
p => (All (x,q)) in { G where G is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = G ) }
let x be bound_QC-variable; :: thesis: for X being Subset of CQC-WFF st p => q in { F where F is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = F ) } & not x in still_not-bound_in p holds
p => (All (x,q)) in { G where G is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = G ) }
let X be Subset of CQC-WFF; :: thesis: ( p => q in { F where F is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = F ) } & not x in still_not-bound_in p implies p => (All (x,q)) in { G where G is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = G ) } )
assume that
A1: p => q in { F where F is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = F ) } and
A2: not x in still_not-bound_in p ; :: thesis: p => (All (x,q)) in { G where G is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = G ) }
ex t being Element of CQC-WFF st
( t = p => q & ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = t ) ) by A1;
then consider f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] such that
A3: f is_a_proof_wrt X and
A4: Effect f = p => q ;
A5: f <> {} by A3, Def5;
reconsider qq = [(p => (All (x,q))),8] as Element of [:CQC-WFF,Proof_Step_Kinds:] by Th43, ZFMISC_1:87;
set h = f ^ <*qq*>;
A6: len (f ^ <*qq*>) = (len f) + (len <*qq*>) by FINSEQ_1:22
.= (len f) + 1 by FINSEQ_1:39 ;
for n being Element of NAT st 1 <= n & n <= len (f ^ <*qq*>) holds
f ^ <*qq*>,n is_a_correct_step_wrt X

proof

let n be Element of NAT ; :: thesis: ( 1 <= n & n <= len (f ^ <*qq*>) implies f ^ <*qq*>,n is_a_correct_step_wrt X )
assume that
A7: 1 <= n and
A8: n <= len (f ^ <*qq*>) ; :: thesis: f ^ <*qq*>,n is_a_correct_step_wrt X

now

per cases ( n <= len f or n = len (f ^ <*qq*>) ) by A6, A8, NAT_1:8;

suppose A9: n <= len f ; :: thesis: f ^ <*qq*>,n is_a_correct_step_wrt X

then f,n is_a_correct_step_wrt X by A3, A7, Def5;
hence f ^ <*qq*>,n is_a_correct_step_wrt X by A7, A9, Th60; :: thesis: verum

end;

suppose A10: n = len (f ^ <*qq*>) ; :: thesis: f ^ <*qq*>,n is_a_correct_step_wrt X

then (f ^ <*qq*>) . n = qq by A6, FINSEQ_1:42;
then A11: ( ((f ^ <*qq*>) . n) `2 = 8 & ((f ^ <*qq*>) . n) `1 = p => (All (x,q)) ) by MCART_1:7;
len f <> 0 by A3, Th58;
then len f in Seg (len f) by FINSEQ_1:3;
then len f in dom f by FINSEQ_1:def 3;
then A12: ((f ^ <*qq*>) . (len f)) `1 = (f . (len f)) `1 by FINSEQ_1:def 7
.= p => q by A4, A5, Def6 ;
A13: 1 <= len f by A3, Th58;
len f < n by A6, A10, NAT_1:13;
hence f ^ <*qq*>,n is_a_correct_step_wrt X by A2, A11, A12, A13, Def4; :: thesis: verum

end;

end;

end;

hence f ^ <*qq*>,n is_a_correct_step_wrt X ; :: thesis: verum

end;

then A14: f ^ <*qq*> is_a_proof_wrt X by Def5;
Effect (f ^ <*qq*>) = ((f ^ <*qq*>) . ((len f) + 1)) `1 by A6, Def6
.= qq `1 by FINSEQ_1:42
.= p => (All (x,q)) by MCART_1:7 ;
hence p => (All (x,q)) in { G where G is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF,Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = G ) } by A14; :: thesis: verum