let A be QC-alphabet ; :: thesis: for t being QC-symbol of A
for p, q being Element of CQC-WFF A
for f being Element of Funcs (,)
for K being Element of Fin holds
( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(),(),()] or [(),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A st [(q '&' r),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A ex u being QC-symbol of A st
( t = u + () & [(r '&' q),u,K,f] in SepQuadruples p ) or ex x being Element of bound_QC-variables A ex u being QC-symbol of A ex h being Element of Funcs (,) st
( u ++ = t & h +* ({x} --> (x. u)) = f & ( [(All (x,q)),u,K,h] in SepQuadruples p or [(All (x,q)),u,(K \ {x}),h] in SepQuadruples p ) ) )

let t be QC-symbol of A; :: thesis: for p, q being Element of CQC-WFF A
for f being Element of Funcs (,)
for K being Element of Fin holds
( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(),(),()] or [(),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A st [(q '&' r),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A ex u being QC-symbol of A st
( t = u + () & [(r '&' q),u,K,f] in SepQuadruples p ) or ex x being Element of bound_QC-variables A ex u being QC-symbol of A ex h being Element of Funcs (,) st
( u ++ = t & h +* ({x} --> (x. u)) = f & ( [(All (x,q)),u,K,h] in SepQuadruples p or [(All (x,q)),u,(K \ {x}),h] in SepQuadruples p ) ) )

let p, q be Element of CQC-WFF A; :: thesis: for f being Element of Funcs (,)
for K being Element of Fin holds
( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(),(),()] or [(),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A st [(q '&' r),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A ex u being QC-symbol of A st
( t = u + () & [(r '&' q),u,K,f] in SepQuadruples p ) or ex x being Element of bound_QC-variables A ex u being QC-symbol of A ex h being Element of Funcs (,) st
( u ++ = t & h +* ({x} --> (x. u)) = f & ( [(All (x,q)),u,K,h] in SepQuadruples p or [(All (x,q)),u,(K \ {x}),h] in SepQuadruples p ) ) )

let f be Element of Funcs (,); :: thesis: for K being Element of Fin holds
( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(),(),()] or [(),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A st [(q '&' r),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A ex u being QC-symbol of A st
( t = u + () & [(r '&' q),u,K,f] in SepQuadruples p ) or ex x being Element of bound_QC-variables A ex u being QC-symbol of A ex h being Element of Funcs (,) st
( u ++ = t & h +* ({x} --> (x. u)) = f & ( [(All (x,q)),u,K,h] in SepQuadruples p or [(All (x,q)),u,(K \ {x}),h] in SepQuadruples p ) ) )

let K be Element of Fin ; :: thesis: ( not [q,t,K,f] in SepQuadruples p or [q,t,K,f] = [p,(),(),()] or [(),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A st [(q '&' r),t,K,f] in SepQuadruples p or ex r being Element of CQC-WFF A ex u being QC-symbol of A st
( t = u + () & [(r '&' q),u,K,f] in SepQuadruples p ) or ex x being Element of bound_QC-variables A ex u being QC-symbol of A ex h being Element of Funcs (,) st
( u ++ = t & h +* ({x} --> (x. u)) = f & ( [(All (x,q)),u,K,h] in SepQuadruples p or [(All (x,q)),u,(K \ {x}),h] in SepQuadruples p ) ) )

assume that
A1: [q,t,K,f] in SepQuadruples p and
A2: [q,t,K,f] <> [p,(),(),()] and
A3: not [(),t,K,f] in SepQuadruples p and
A4: for r being Element of CQC-WFF A holds not [(q '&' r),t,K,f] in SepQuadruples p and
A5: for r being Element of CQC-WFF A
for u being QC-symbol of A holds
( not t = u + () or not [(r '&' q),u,K,f] in SepQuadruples p ) and
A6: for x being Element of bound_QC-variables A
for u being QC-symbol of A
for h being Element of Funcs (,) holds
( not u ++ = t or not h +* ({x} --> (x. u)) = f or ( not [(All (x,q)),u,K,h] in SepQuadruples p & not [(All (x,q)),u,(K \ {x}),h] in SepQuadruples p ) ) ; :: thesis: contradiction
reconsider Y = () \ {[q,t,K,f]} as Subset of [:(),(),(),(Funcs (,)):] ;
A7: SepQuadruples p is_Sep-closed_on p by Def13;
A8: for q being Element of CQC-WFF A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(),t,K,f] in Y holds
[q,t,K,f] in Y
proof
let s be Element of CQC-WFF A; :: thesis: for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(),t,K,f] in Y holds
[s,t,K,f] in Y

let u be QC-symbol of A; :: thesis: for K being Element of Fin
for f being Element of Funcs (,) st [(),u,K,f] in Y holds
[s,u,K,f] in Y

let L be Element of Fin ; :: thesis: for f being Element of Funcs (,) st [(),u,L,f] in Y holds
[s,u,L,f] in Y

let h be Element of Funcs (,); :: thesis: ( [(),u,L,h] in Y implies [s,u,L,h] in Y )
assume A9: [(),u,L,h] in Y ; :: thesis: [s,u,L,h] in Y
then ( s <> q or u <> t or L <> K or f <> h ) by ;
then A10: [s,u,L,h] <> [q,t,K,f] by XTUPLE_0:5;
[(),u,L,h] in SepQuadruples p by ;
then [s,u,L,h] in SepQuadruples p by A7;
hence [s,u,L,h] in Y by ; :: thesis: verum
end;
A11: for q, r being Element of CQC-WFF A
for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(q '&' r),t,K,f] in Y holds
( [q,t,K,f] in Y & [r,(t + ()),K,f] in Y )
proof
let s, r be Element of CQC-WFF A; :: thesis: for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(s '&' r),t,K,f] in Y holds
( [s,t,K,f] in Y & [r,(t + ()),K,f] in Y )

let u be QC-symbol of A; :: thesis: for K being Element of Fin
for f being Element of Funcs (,) st [(s '&' r),u,K,f] in Y holds
( [s,u,K,f] in Y & [r,(u + ()),K,f] in Y )

let L be Element of Fin ; :: thesis: for f being Element of Funcs (,) st [(s '&' r),u,L,f] in Y holds
( [s,u,L,f] in Y & [r,(u + ()),L,f] in Y )

let h be Element of Funcs (,); :: thesis: ( [(s '&' r),u,L,h] in Y implies ( [s,u,L,h] in Y & [r,(u + ()),L,h] in Y ) )
assume [(s '&' r),u,L,h] in Y ; :: thesis: ( [s,u,L,h] in Y & [r,(u + ()),L,h] in Y )
then A12: [(s '&' r),u,L,h] in SepQuadruples p by XBOOLE_0:def 5;
then ( s <> q or u <> t or L <> K or f <> h ) by A4;
then A13: [s,u,L,h] <> [q,t,K,f] by XTUPLE_0:5;
[s,u,L,h] in SepQuadruples p by ;
hence [s,u,L,h] in Y by ; :: thesis: [r,(u + ()),L,h] in Y
( r <> q or L <> K or f <> h or u + () <> t ) by ;
then A14: [r,(u + ()),L,h] <> [q,t,K,f] by XTUPLE_0:5;
[r,(u + ()),L,h] in SepQuadruples p by ;
hence [r,(u + ()),L,h] in Y by ; :: thesis: verum
end;
A15: Y c= SepQuadruples p by XBOOLE_1:36;
A16: 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
for f being Element of Funcs (,) st [(All (x,q)),t,K,f] in Y holds
[q,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in Y
proof
let s 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
for f being Element of Funcs (,) st [(All (x,s)),t,K,f] in Y holds
[s,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in Y

let x be Element of bound_QC-variables A; :: thesis: for t being QC-symbol of A
for K being Element of Fin
for f being Element of Funcs (,) st [(All (x,s)),t,K,f] in Y holds
[s,(t ++),(K \/ {x}),(f +* (x .--> (x. t)))] in Y

let u be QC-symbol of A; :: thesis: for K being Element of Fin
for f being Element of Funcs (,) st [(All (x,s)),u,K,f] in Y holds
[s,(u ++),(K \/ {x}),(f +* (x .--> (x. u)))] in Y

let L be Element of Fin ; :: thesis: for f being Element of Funcs (,) st [(All (x,s)),u,L,f] in Y holds
[s,(u ++),(L \/ {x}),(f +* (x .--> (x. u)))] in Y

let h be Element of Funcs (,); :: thesis: ( [(All (x,s)),u,L,h] in Y implies [s,(u ++),(L \/ {x}),(h +* (x .--> (x. u)))] in Y )
assume A17: [(All (x,s)),u,L,h] in Y ; :: thesis: [s,(u ++),(L \/ {x}),(h +* (x .--> (x. u)))] in Y
now :: thesis: ( not [(All (x,q)),u,K,h] in SepQuadruples p & not [(All (x,q)),u,(K \ {x}),h] in SepQuadruples p & s = q implies not L \/ {x} = K )
assume that
A18: not [(All (x,q)),u,K,h] in SepQuadruples p and
A19: not [(All (x,q)),u,(K \ {x}),h] in SepQuadruples p ; :: thesis: ( s = q implies not L \/ {x} = K )
A20: ( s <> q or ( L <> K & L <> K \ {x} ) ) by ;
assume A21: s = q ; :: thesis: not L \/ {x} = K
assume A22: L \/ {x} = K ; :: thesis: contradiction
then K \ {x} = L \ {x} by XBOOLE_1:40;
hence contradiction by A20, A21, A22, ZFMISC_1:40, ZFMISC_1:57; :: thesis: verum
end;
then ( s <> q or u ++ <> t or L \/ {x} <> K or f <> h +* ({x} --> (x. u)) ) by A6;
then A23: [s,(u ++),(L \/ {x}),(h +* (x .--> (x. u)))] <> [q,t,K,f] by XTUPLE_0:5;
[(All (x,s)),u,L,h] in SepQuadruples p by ;
then [s,(u ++),(L \/ {x}),(h +* (x .--> (x. u)))] in SepQuadruples p by A7;
hence [s,(u ++),(L \/ {x}),(h +* (x .--> (x. u)))] in Y by ; :: thesis: verum
end;
[p,(),(),()] in SepQuadruples p by A7;
then [p,(),(),()] in Y by ;
then Y is_Sep-closed_on p by A8, A11, A16;
then SepQuadruples p c= Y by Def13;
then Y = SepQuadruples p by A15;
hence contradiction by A1, ZFMISC_1:57; :: thesis: verum