let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InputVertices S1 misses InnerVertices S2 & S = S1 +* S2 implies for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2
for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for n being natural Number holds (Following (s,n)) | the carrier of S1 = Following (s1,n) )
assume A1: ( InputVertices S1 misses InnerVertices S2 & S = S1 +* S2 ) ; :: thesis: for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2
for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for n being natural Number holds (Following (s,n)) | the carrier of S1 = Following (s1,n)
let A1 be non-empty Circuit of S1; :: thesis: for A2 being non-empty Circuit of S2
for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for n being natural Number holds (Following (s,n)) | the carrier of S1 = Following (s1,n)
let A2 be non-empty Circuit of S2; :: thesis: for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for n being natural Number holds (Following (s,n)) | the carrier of S1 = Following (s1,n)
let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for n being natural Number holds (Following (s,n)) | the carrier of S1 = Following (s1,n) )
assume A2: ( A1 tolerates A2 & A = A1 +* A2 ) ; :: thesis: for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for n being natural Number holds (Following (s,n)) | the carrier of S1 = Following (s1,n)
let s be State of A; :: thesis: for s1 being State of A1 st s1 = s | the carrier of S1 holds
for n being natural Number 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 natural Number holds (Following (s,n)) | the carrier of S1 = Following (s1,n) )
assume A3: s1 = s | the carrier of S1 ; :: thesis: for n being natural Number holds (Following (s,n)) | the carrier of S1 = Following (s1,n)
let n be natural Number ; :: thesis: (Following (s,n)) | the carrier of S1 = Following (s1,n)
A0: n is Nat by TARSKI:1;
defpred S₃[ Nat] means (Following (s,$1)) | the carrier of S1 = Following (s1,$1);

A4: now :: thesis: for n being Nat st S₃[n] holds
S₃[n + 1]

let n be Nat; :: thesis: ( S₃[n] implies S₃[n + 1] )
A5: ( Following (s,(n + 1)) = Following (Following (s,n)) & Following (Following (s1,n)) = Following (s1,(n + 1)) ) by FACIRC_1:12;
assume S₃[n] ; :: thesis: S₃[n + 1]
hence S₃[n + 1] by A1, A2, A5, Th10; :: thesis: verum

end;

(Following (s,0)) | the carrier of S1 = s1 by A3, FACIRC_1:11
.= Following (s1,0) by FACIRC_1:11 ;
then A6: S₃[ 0 ] ;
for n being Nat holds S₃[n] from NAT_1:sch 2(A6, A4);
hence (Following (s,n)) | the carrier of S1 = Following (s1,n) by A0; :: thesis: verum