let S1, S2 be non empty non void unsplit gate`1=arity gate`2isBoolean ManySortedSign ; :: thesis: ( InnerVertices S2 misses InputVertices S1 implies for A1 being gate`2=den Boolean Circuit of S1

for A2 being gate`2=den Boolean Circuit of S2

for s being State of (A1 +* A2)

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n) )

assume A1: InnerVertices S2 misses InputVertices S1 ; :: thesis: for A1 being gate`2=den Boolean Circuit of S1

for A2 being gate`2=den Boolean Circuit of S2

for s being State of (A1 +* A2)

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n)

let A1 be gate`2=den Boolean Circuit of S1; :: thesis: for A2 being gate`2=den Boolean Circuit of S2

for s being State of (A1 +* A2)

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n)

let A2 be gate`2=den Boolean Circuit of S2; :: thesis: for s being State of (A1 +* A2)

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n)

let s be State of (A1 +* A2); :: thesis: for s1 being State of A1 st s1 = s | the carrier of S1 holds

for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n)

let s1 be State of A1; :: thesis: ( s1 = s | the carrier of S1 implies for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n) )

assume A2: s1 = s | the carrier of S1 ; :: thesis: for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n)

defpred S_{1}[ Nat] means (Following (s,$1)) | the carrier of S1 = Following (s1,$1);

A3: for n being Nat st S_{1}[n] holds

S_{1}[n + 1]

then A5: S_{1}[ 0 ]
by A2, Th11;

thus for n being Nat holds S_{1}[n]
from NAT_1:sch 2(A5, A3); :: thesis: verum

for A2 being gate`2=den Boolean Circuit of S2

for s being State of (A1 +* A2)

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n) )

assume A1: InnerVertices S2 misses InputVertices S1 ; :: thesis: for A1 being gate`2=den Boolean Circuit of S1

for A2 being gate`2=den Boolean Circuit of S2

for s being State of (A1 +* A2)

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n)

let A1 be gate`2=den Boolean Circuit of S1; :: thesis: for A2 being gate`2=den Boolean Circuit of S2

for s being State of (A1 +* A2)

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n)

let A2 be gate`2=den Boolean Circuit of S2; :: thesis: for s being State of (A1 +* A2)

for s1 being State of A1 st s1 = s | the carrier of S1 holds

for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n)

let s be State of (A1 +* A2); :: thesis: for s1 being State of A1 st s1 = s | the carrier of S1 holds

for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n)

let s1 be State of A1; :: thesis: ( s1 = s | the carrier of S1 implies for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n) )

assume A2: s1 = s | the carrier of S1 ; :: thesis: for n being Nat holds (Following (s,n)) | the carrier of S1 = Following (s1,n)

defpred S

A3: for n being Nat st S

S

proof

Following (s,0) = s
by Th11;
let n be Nat; :: thesis: ( S_{1}[n] implies S_{1}[n + 1] )

assume A4: (Following (s,n)) | the carrier of S1 = Following (s1,n) ; :: thesis: S_{1}[n + 1]

thus (Following (s,(n + 1))) | the carrier of S1 = (Following (Following (s,n))) | the carrier of S1 by Th12

.= Following (Following (s1,n)) by A1, A4, Th28

.= Following (s1,(n + 1)) by Th12 ; :: thesis: verum

end;assume A4: (Following (s,n)) | the carrier of S1 = Following (s1,n) ; :: thesis: S

thus (Following (s,(n + 1))) | the carrier of S1 = (Following (Following (s,n))) | the carrier of S1 by Th12

.= Following (Following (s1,n)) by A1, A4, Th28

.= Following (s1,(n + 1)) by Th12 ; :: thesis: verum

then A5: S

thus for n being Nat holds S