let A be QC-alphabet ; :: thesis: for t, u being QC-symbol of A
for p, q being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Element of Fin (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: K holds
u < t
let t, u be QC-symbol of A; :: thesis: for p, q being Element of CQC-WFF A
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Element of Fin (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: K holds
u < t
let p, q be Element of CQC-WFF A; :: thesis: for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A))
for K being Element of Fin (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: K holds
u < t
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); :: thesis: for K being Element of Fin (bound_QC-variables A) st [q,t,K,f] in SepQuadruples p & x. u in f .: K holds
u < t
let K be Element of Fin (bound_QC-variables A); :: thesis: ( [q,t,K,f] in SepQuadruples p & x. u in f .: K implies u < t )
defpred S₁[ Element of CQC-WFF A, QC-symbol of A, Element of Fin (bound_QC-variables A), Function] means for u being QC-symbol of A st x. u in $4 .: $3 holds
u < $2;
A1: for q being Element of CQC-WFF A
for v being QC-symbol of A
for K being Element of Fin (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [('not' q),v,K,f] in SepQuadruples p & S₁[ 'not' q,v,K,f] holds
S₁[q,v,K,f] ;

A2: now :: thesis: for q, r being Element of CQC-WFF A
for v being QC-symbol of A
for K being Element of Fin (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(q '&' r),v,K,f] in SepQuadruples p & S₁[q '&' r,v,K,f] holds
( S₁[q,v,K,f] & S₁[r,v + (QuantNbr q),K,f] )

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

proof

let u be QC-symbol of A; :: thesis: ( x. u in f .: K implies u < v + (QuantNbr q) )
A4: v <= v + (QuantNbr q) by QC_LANG1:31;
assume x. u in f .: K ; :: thesis: u < v + (QuantNbr q)
hence u < v + (QuantNbr q) by A3, A4, QC_LANG1:30; :: thesis: verum

end;

end;

A5: now :: thesis: for q being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for v being QC-symbol of A
for K being Element of Fin (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [(All (x,q)),v,K,f] in SepQuadruples p & S₁[ All (x,q),v,K,f] holds
S₁[q,v ++ ,K \/ {.x.},f +* (x .--> (x. v))]

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

proof

let u be QC-symbol of A; :: thesis: ( x. u in (f +* (x .--> (x. v))) .: (K \/ {.x.}) implies u < v ++ )
assume x. u in (f +* (x .--> (x. v))) .: (K \/ {x}) ; :: thesis: u < v ++
then x. u in ((f +* (x .--> (x. v))) .: K) \/ ((f +* (x .--> (x. v))) .: {x}) by RELAT_1:120;
then A7: ( x. u in (f +* (x .--> (x. v))) .: K or x. u in Im ((f +* (x .--> (x. v))),x) ) by XBOOLE_0:def 3;
(f +* (x .--> (x. v))) .: K c= (f .: K) \/ {(x. v)} by Th2;
then ( x. u in f .: K or x. u in {(x. v)} ) by A7, Th1, XBOOLE_0:def 3;
then ( u < v or x. u = x. v ) by A6, TARSKI:def 1;
then ( u < v or u = v ) by XTUPLE_0:1;
then ( u <= v & v < v ++ ) by QC_LANG1:22, QC_LANG1:27, QC_LANG1:def 34;
hence u < v ++ by QC_LANG1:29; :: thesis: verum

end;

end;

A8: S₁[p, index p, {}. (bound_QC-variables A), id (bound_QC-variables A)] ;
for q being Element of CQC-WFF A
for v being QC-symbol of A
for K being Element of Fin (bound_QC-variables A)
for f being Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)) st [q,v,K,f] in SepQuadruples p holds
S₁[q,v,K,f] from CQC_SIM1:sch 6(A8, A1, A2, A5);
hence ( [q,t,K,f] in SepQuadruples p & x. u in f .: K implies u < t ) ; :: thesis: verum