let S1, S2 be non empty non void unsplit gate`1=arity gate`2isBoolean ManySortedSign ; :: thesis: ( InnerVertices S1 misses InputVertices S2 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 s2 being State of A2 st s2 = s | the carrier of S2 holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n )
assume A1: InnerVertices S1 misses InputVertices S2 ; :: 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 s2 being State of A2 st s2 = s | the carrier of S2 holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,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 s2 being State of A2 st s2 = s | the carrier of S2 holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n
let A2 be gate`2=den Boolean Circuit of S2; :: thesis: for s being State of (A1 +* A2)
for s2 being State of A2 st s2 = s | the carrier of S2 holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n
let s be State of (A1 +* A2); :: thesis: for s2 being State of A2 st s2 = s | the carrier of S2 holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n
let s2 be State of A2; :: thesis: ( s2 = s | the carrier of S2 implies for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n )
assume A2: s2 = s | the carrier of S2 ; :: thesis: for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n
defpred S₁[ Nat] means (Following s,$1) | the carrier of S2 = Following s2,$1;
A3: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A4: (Following s,n) | the carrier of S2 = Following s2,n ; :: thesis: S₁[n + 1]
thus (Following s,(n + 1)) | the carrier of S2 = (Following (Following s,n)) | the carrier of S2 by Th12
.= Following (Following s2,n) by A1, A4, Th29
.= Following s2,(n + 1) by Th12 ; :: thesis: verum

end;

Following s,0 = s by Th11;
then A5: S₁[ 0 ] by A2, Th11;
thus for n being Nat holds S₁[n] from NAT_1:sch 2(A5, A3); :: thesis: verum