let A be QC-alphabet ; :: thesis: for p, q being Element of CQC-WFF A
for t 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,t,K,f] in SepQuadruples p holds
still_not-bound_in q c= (still_not-bound_in p) \/ K
let p be Element of CQC-WFF A; :: thesis: for q being Element of CQC-WFF A
for t 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,t,K,f] in SepQuadruples p holds
still_not-bound_in q c= (still_not-bound_in p) \/ K
deffunc H₅( QC-formula of A) -> Element of bool (bound_QC-variables A) = still_not-bound_in $1;
defpred S₁[ QC-formula of A, set , set , set ] means H₅($1) c= H₅(p) \/ $3;
A1: for q being Element of CQC-WFF A
for t 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),t,K,f] in SepQuadruples p & S₁[ 'not' q,t,K,f] holds
S₁[q,t,K,f] by QC_LANG3:7;

A2: now :: thesis: for q, r being Element of CQC-WFF A
for t 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),t,K,f] in SepQuadruples p & S₁[q '&' r,t,K,f] holds
( S₁[q,t,K,f] & S₁[r,t + (QuantNbr q),K,f] )

let q, r be Element of CQC-WFF A; :: thesis: for t 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),t,K,f] in SepQuadruples p & S₁[q '&' r,t,K,f] holds
( S₁[q,t,K,f] & S₁[r,t + (QuantNbr q),K,f] )
let t 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),t,K,f] in SepQuadruples p & S₁[q '&' r,t,K,f] holds
( S₁[q,t,K,f] & S₁[r,t + (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),t,K,f] in SepQuadruples p & S₁[q '&' r,t,K,f] holds
( S₁[q,t,K,f] & S₁[r,t + (QuantNbr q),K,f] )
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); :: thesis: ( [(q '&' r),t,K,f] in SepQuadruples p & S₁[q '&' r,t,K,f] implies ( S₁[q,t,K,f] & S₁[r,t + (QuantNbr q),K,f] ) )
assume that
[(q '&' r),t,K,f] in SepQuadruples p and
A3: S₁[q '&' r,t,K,f] ; :: thesis: ( S₁[q,t,K,f] & S₁[r,t + (QuantNbr q),K,f] )
A4: still_not-bound_in (q '&' r) = (still_not-bound_in q) \/ (still_not-bound_in r) by QC_LANG3:10;
then A5: still_not-bound_in r c= still_not-bound_in (q '&' r) by XBOOLE_1:7;
still_not-bound_in q c= still_not-bound_in (q '&' r) by A4, XBOOLE_1:7;
hence ( S₁[q,t,K,f] & S₁[r,t + (QuantNbr q),K,f] ) by A3, A5, XBOOLE_1:1; :: thesis: verum

end;

A6: now :: thesis: for q being Element of CQC-WFF A
for x being Element of bound_QC-variables A
for t 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)),t,K,f] in SepQuadruples p & S₁[ All (x,q),t,K,f] holds
S₁[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]

let q be Element of CQC-WFF A; :: thesis: for x being Element of bound_QC-variables A
for t 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)),t,K,f] in SepQuadruples p & S₁[ All (x,q),t,K,f] holds
S₁[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
let x be Element of bound_QC-variables A; :: thesis: for t 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)),t,K,f] in SepQuadruples p & S₁[ All (x,q),t,K,f] holds
S₁[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
let t 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)),t,K,f] in SepQuadruples p & S₁[ All (x,q),t,K,f] holds
S₁[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
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)),t,K,f] in SepQuadruples p & S₁[ All (x,q),t,K,f] holds
S₁[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
let f be Element of Funcs ((bound_QC-variables A),(bound_QC-variables A)); :: thesis: ( [(All (x,q)),t,K,f] in SepQuadruples p & S₁[ All (x,q),t,K,f] implies S₁[q,t ++ ,K \/ {x},f +* (x .--> (x. t))] )
assume that
[(All (x,q)),t,K,f] in SepQuadruples p and
A7: S₁[ All (x,q),t,K,f] ; :: thesis: S₁[q,t ++ ,K \/ {x},f +* (x .--> (x. t))]
still_not-bound_in (All (x,q)) = (still_not-bound_in q) \ {x} by QC_LANG3:12;
then still_not-bound_in q c= ((still_not-bound_in p) \/ K) \/ {x} by A7, XBOOLE_1:44;
hence S₁[q,t ++ ,K \/ {x},f +* (x .--> (x. t))] by XBOOLE_1:4; :: thesis: verum

end;

A8: S₁[p, index p, {}. (bound_QC-variables A), id (bound_QC-variables A)] ;
thus for q being Element of CQC-WFF A
for t 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,t,K,f] in SepQuadruples p holds
S₁[q,t,K,f] from CQC_SIM1:sch 6(A8, A1, A2, A6); :: thesis: verum