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 S₁[ Nat] means (Following (s,$1)) | the carrier of S1 = Following (s1,$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 S1 = Following (s1,n) ; :: thesis: S₁[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;

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