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 & Effect f = p => q ) ;
A4: f <> {} by A3, Def5;
reconsider qq = [(p => (All x,q)),8] as Element of [:CQC-WFF ,Proof_Step_Kinds :] by Th43, ZFMISC_1:106;
set h = f ^ <*qq*>;
A5: len (f ^ <*qq*>) = (len f) + (len <*qq*>) by FINSEQ_1:35
.= (len f) + 1 by FINSEQ_1:56 ;
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 A6: ( 1 <= n & 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 A5, A6, NAT_1:8;

suppose A7: n <= len f ; :: thesis: f ^ <*qq*>,n is_a_correct_step_wrt X

then f,n is_a_correct_step_wrt X by A3, A6, Def5;
hence f ^ <*qq*>,n is_a_correct_step_wrt X by A6, A7, Th60; :: thesis: verum

end;

suppose A8: n = len (f ^ <*qq*>) ; :: thesis: f ^ <*qq*>,n is_a_correct_step_wrt X

then (f ^ <*qq*>) . n = qq by A5, FINSEQ_1:59;
then A9: ( ((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_1:5;
then len f in dom f by FINSEQ_1:def 3;
then A10: ((f ^ <*qq*>) . (len f)) `1 = (f . (len f)) `1 by FINSEQ_1:def 7
.= p => q by A3, A4, Def6 ;
( 1 <= len f & len f < n ) by A3, A5, A8, Th58, NAT_1:13;
hence f ^ <*qq*>,n is_a_correct_step_wrt X by A2, A9, A10, Def4; :: thesis: verum

end;

end;

end;

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

end;

then A11: f ^ <*qq*> is_a_proof_wrt X by Def5;
Effect (f ^ <*qq*>) = ((f ^ <*qq*>) . ((len f) + 1)) `1 by A5, Def6
.= qq `1 by FINSEQ_1:59
.= 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 A11; :: thesis: verum