let S be non empty non void unsplit gate`1=arity gate`2isBoolean ManySortedSign ; :: thesis: for A being non-empty gate`2=den Boolean Circuit of S

for s being State of A

for p being FinSequence

for f being Function st [p,f] in the carrier' of S holds

(Following s) . [p,f] = f . (s * p)

let A be non-empty gate`2=den Boolean Circuit of S; :: thesis: for s being State of A

for p being FinSequence

for f being Function st [p,f] in the carrier' of S holds

(Following s) . [p,f] = f . (s * p)

let s be State of A; :: thesis: for p being FinSequence

for f being Function st [p,f] in the carrier' of S holds

(Following s) . [p,f] = f . (s * p)

let p be FinSequence; :: thesis: for f being Function st [p,f] in the carrier' of S holds

(Following s) . [p,f] = f . (s * p)

let f be Function; :: thesis: ( [p,f] in the carrier' of S implies (Following s) . [p,f] = f . (s * p) )

assume [p,f] in the carrier' of S ; :: thesis: (Following s) . [p,f] = f . (s * p)

then reconsider g = [p,f] as Gate of S ;

A1: ( g `1 = p & g `2 = f ) ;

A2: the_result_sort_of g = the ResultSort of S . g by MSUALG_1:def 2

.= g by CIRCCOMB:44 ;

the_arity_of g = the Arity of S . g by MSUALG_1:def 1

.= [( the Arity of S . g),(g `2)] `1

.= g `1 by CIRCCOMB:def 8 ;

hence (Following s) . [p,f] = f . (s * p) by A1, A2, Th34; :: thesis: verum

for s being State of A

for p being FinSequence

for f being Function st [p,f] in the carrier' of S holds

(Following s) . [p,f] = f . (s * p)

let A be non-empty gate`2=den Boolean Circuit of S; :: thesis: for s being State of A

for p being FinSequence

for f being Function st [p,f] in the carrier' of S holds

(Following s) . [p,f] = f . (s * p)

let s be State of A; :: thesis: for p being FinSequence

for f being Function st [p,f] in the carrier' of S holds

(Following s) . [p,f] = f . (s * p)

let p be FinSequence; :: thesis: for f being Function st [p,f] in the carrier' of S holds

(Following s) . [p,f] = f . (s * p)

let f be Function; :: thesis: ( [p,f] in the carrier' of S implies (Following s) . [p,f] = f . (s * p) )

assume [p,f] in the carrier' of S ; :: thesis: (Following s) . [p,f] = f . (s * p)

then reconsider g = [p,f] as Gate of S ;

A1: ( g `1 = p & g `2 = f ) ;

A2: the_result_sort_of g = the ResultSort of S . g by MSUALG_1:def 2

.= g by CIRCCOMB:44 ;

the_arity_of g = the Arity of S . g by MSUALG_1:def 1

.= [( the Arity of S . g),(g `2)] `1

.= g `1 by CIRCCOMB:def 8 ;

hence (Following s) . [p,f] = f . (s * p) by A1, A2, Th34; :: thesis: verum