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 v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v )

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 v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

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 v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

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 v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

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

for v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

let s1 be State of A1; :: thesis: ( s1 = s | the carrier of S1 implies for v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v )

assume A2: s1 = s | the carrier of S1 ; :: thesis: for v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

let v be set ; :: thesis: ( v in the carrier of S1 implies for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v )

assume A3: v in the carrier of S1 ; :: thesis: for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

let n be Nat; :: thesis: (Following (s,n)) . v = (Following (s1,n)) . v

A4: the carrier of S1 = dom (Following (s1,n)) by CIRCUIT1:3;

(Following (s,n)) | the carrier of S1 = Following (s1,n) by A1, A2, Th30;

hence (Following (s,n)) . v = (Following (s1,n)) . v by A3, A4, FUNCT_1:47; :: 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 v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v )

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 v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

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 v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

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 v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

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

for v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

let s1 be State of A1; :: thesis: ( s1 = s | the carrier of S1 implies for v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v )

assume A2: s1 = s | the carrier of S1 ; :: thesis: for v being set st v in the carrier of S1 holds

for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

let v be set ; :: thesis: ( v in the carrier of S1 implies for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v )

assume A3: v in the carrier of S1 ; :: thesis: for n being Nat holds (Following (s,n)) . v = (Following (s1,n)) . v

let n be Nat; :: thesis: (Following (s,n)) . v = (Following (s1,n)) . v

A4: the carrier of S1 = dom (Following (s1,n)) by CIRCUIT1:3;

(Following (s,n)) | the carrier of S1 = Following (s1,n) by A1, A2, Th30;

hence (Following (s,n)) . v = (Following (s1,n)) . v by A3, A4, FUNCT_1:47; :: thesis: verum