let q, p be Element of CQC-WFF ; :: thesis: for m, i being Element of NAT
for f being Element of Funcs (bound_QC-variables,bound_QC-variables)
for K being Finite_Subset of bound_QC-variables st [q,m,K,f] in SepQuadruples p & x. i in f .: K holds
i < m
let m, i be Element of NAT ; :: thesis: for f being Element of Funcs (bound_QC-variables,bound_QC-variables)
for K being Finite_Subset of bound_QC-variables st [q,m,K,f] in SepQuadruples p & x. i in f .: K holds
i < m
let f be Element of Funcs (bound_QC-variables,bound_QC-variables); :: thesis: for K being Finite_Subset of bound_QC-variables st [q,m,K,f] in SepQuadruples p & x. i in f .: K holds
i < m
let K be Finite_Subset of bound_QC-variables; :: thesis: ( [q,m,K,f] in SepQuadruples p & x. i in f .: K implies i < m )
defpred S₁[ Element of CQC-WFF , Element of NAT , Finite_Subset of bound_QC-variables, Function] means for i being Element of NAT st x. i in $4 .: $3 holds
i < $2;
A1: for q being Element of CQC-WFF
for k being Element of NAT
for K being Finite_Subset of bound_QC-variables
for f being Element of Funcs (bound_QC-variables,bound_QC-variables) st [('not' q),k,K,f] in SepQuadruples p & S₁[ 'not' q,k,K,f] holds
S₁[q,k,K,f] ;

A2: now

let q, r be Element of CQC-WFF ; :: thesis: for k being Element of NAT
for K being Finite_Subset of bound_QC-variables
for f being Element of Funcs (bound_QC-variables,bound_QC-variables) st [(q '&' r),k,K,f] in SepQuadruples p & S₁[q '&' r,k,K,f] holds
( S₁[q,k,K,f] & S₁[r,k + (QuantNbr q),K,f] )
let k be Element of NAT ; :: thesis: for K being Finite_Subset of bound_QC-variables
for f being Element of Funcs (bound_QC-variables,bound_QC-variables) st [(q '&' r),k,K,f] in SepQuadruples p & S₁[q '&' r,k,K,f] holds
( S₁[q,k,K,f] & S₁[r,k + (QuantNbr q),K,f] )
let K be Finite_Subset of bound_QC-variables; :: thesis: for f being Element of Funcs (bound_QC-variables,bound_QC-variables) st [(q '&' r),k,K,f] in SepQuadruples p & S₁[q '&' r,k,K,f] holds
( S₁[q,k,K,f] & S₁[r,k + (QuantNbr q),K,f] )
let f be Element of Funcs (bound_QC-variables,bound_QC-variables); :: thesis: ( [(q '&' r),k,K,f] in SepQuadruples p & S₁[q '&' r,k,K,f] implies ( S₁[q,k,K,f] & S₁[r,k + (QuantNbr q),K,f] ) )
assume [(q '&' r),k,K,f] in SepQuadruples p ; :: thesis: ( S₁[q '&' r,k,K,f] implies ( S₁[q,k,K,f] & S₁[r,k + (QuantNbr q),K,f] ) )
assume A3: S₁[q '&' r,k,K,f] ; :: thesis: ( S₁[q,k,K,f] & S₁[r,k + (QuantNbr q),K,f] )
hence S₁[q,k,K,f] ; :: thesis: S₁[r,k + (QuantNbr q),K,f]
thus S₁[r,k + (QuantNbr q),K,f] :: thesis: verum

proof

let i be Element of NAT ; :: thesis: ( x. i in f .: K implies i < k + (QuantNbr q) )
A4: k <= k + (QuantNbr q) by NAT_1:11;
assume x. i in f .: K ; :: thesis: i < k + (QuantNbr q)
hence i < k + (QuantNbr q) by A3, A4, XXREAL_0:2; :: thesis: verum

end;

end;

A5: now

let q be Element of CQC-WFF ; :: thesis: for x being Element of bound_QC-variables
for k being Element of NAT
for K being Finite_Subset of bound_QC-variables
for f being Element of Funcs (bound_QC-variables,bound_QC-variables) st [(All (x,q)),k,K,f] in SepQuadruples p & S₁[ All (x,q),k,K,f] holds
S₁[q,k + 1,K \/ {.x.},f +* (x .--> (x. k))]
let x be Element of bound_QC-variables ; :: thesis: for k being Element of NAT
for K being Finite_Subset of bound_QC-variables
for f being Element of Funcs (bound_QC-variables,bound_QC-variables) st [(All (x,q)),k,K,f] in SepQuadruples p & S₁[ All (x,q),k,K,f] holds
S₁[q,k + 1,K \/ {.x.},f +* (x .--> (x. k))]
let k be Element of NAT ; :: thesis: for K being Finite_Subset of bound_QC-variables
for f being Element of Funcs (bound_QC-variables,bound_QC-variables) st [(All (x,q)),k,K,f] in SepQuadruples p & S₁[ All (x,q),k,K,f] holds
S₁[q,k + 1,K \/ {.x.},f +* (x .--> (x. k))]
let K be Finite_Subset of bound_QC-variables; :: thesis: for f being Element of Funcs (bound_QC-variables,bound_QC-variables) st [(All (x,q)),k,K,f] in SepQuadruples p & S₁[ All (x,q),k,K,f] holds
S₁[q,k + 1,K \/ {.x.},f +* (x .--> (x. k))]
let f be Element of Funcs (bound_QC-variables,bound_QC-variables); :: thesis: ( [(All (x,q)),k,K,f] in SepQuadruples p & S₁[ All (x,q),k,K,f] implies S₁[q,k + 1,K \/ {.x.},f +* (x .--> (x. k))] )
assume [(All (x,q)),k,K,f] in SepQuadruples p ; :: thesis: ( S₁[ All (x,q),k,K,f] implies S₁[q,k + 1,K \/ {.x.},f +* (x .--> (x. k))] )
assume A6: S₁[ All (x,q),k,K,f] ; :: thesis: S₁[q,k + 1,K \/ {.x.},f +* (x .--> (x. k))]
thus S₁[q,k + 1,K \/ {.x.},f +* (x .--> (x. k))] :: thesis: verum

proof

let i be Element of NAT ; :: thesis: ( x. i in (f +* (x .--> (x. k))) .: (K \/ {.x.}) implies i < k + 1 )
assume x. i in (f +* (x .--> (x. k))) .: (K \/ {x}) ; :: thesis: i < k + 1
then x. i in ((f +* (x .--> (x. k))) .: K) \/ ((f +* (x .--> (x. k))) .: {x}) by RELAT_1:120;
then A7: ( x. i in (f +* (x .--> (x. k))) .: K or x. i in Im ((f +* (x .--> (x. k))),x) ) by XBOOLE_0:def 3;
(f +* (x .--> (x. k))) .: K c= (f .: K) \/ {(x. k)} by Th2;
then ( x. i in f .: K or x. i in {(x. k)} ) by A7, Th1, XBOOLE_0:def 3;
then ( i < k or x. i = x. k ) by A6, TARSKI:def 1;
then i <= k by ZFMISC_1:27;
hence i < k + 1 by NAT_1:13; :: thesis: verum

end;

end;

A8: S₁[p, index p, {}. bound_QC-variables, id bound_QC-variables] by RELAT_1:116;
for q being Element of CQC-WFF
for k being Element of NAT
for K being Finite_Subset of bound_QC-variables
for f being Element of Funcs (bound_QC-variables,bound_QC-variables) st [q,k,K,f] in SepQuadruples p holds
S₁[q,k,K,f] from CQC_SIM1:sch 6(A8, A1, A2, A5);
hence ( [q,m,K,f] in SepQuadruples p & x. i in f .: K implies i < m ) ; :: thesis: verum