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 S_{3}[ Nat] means (Following (s,$1)) | the carrier of S1 = Following (s1,$1);

.= Following (s1,0) by FACIRC_1:11 ;

then A6: S_{3}[ 0 ]
;

for n being Nat holds S_{3}[n]
from NAT_1:sch 2(A6, A4);

hence (Following (s,n)) | the carrier of S1 = Following (s1,n) by A0; :: thesis: verum

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 S

A4: now :: thesis: for n being Nat st S_{3}[n] holds

S_{3}[n + 1]

(Following (s,0)) | the carrier of S1 =
s1
by A3, FACIRC_1:11
S

let n be Nat; :: thesis: ( S_{3}[n] implies S_{3}[n + 1] )

A5: ( Following (s,(n + 1)) = Following (Following (s,n)) & Following (Following (s1,n)) = Following (s1,(n + 1)) ) by FACIRC_1:12;

assume S_{3}[n]
; :: thesis: S_{3}[n + 1]

hence S_{3}[n + 1]
by A1, A2, A5, Th10; :: thesis: verum

end;A5: ( Following (s,(n + 1)) = Following (Following (s,n)) & Following (Following (s1,n)) = Following (s1,(n + 1)) ) by FACIRC_1:12;

assume S

hence S

.= Following (s1,0) by FACIRC_1:11 ;

then A6: S

for n being Nat holds S

hence (Following (s,n)) | the carrier of S1 = Following (s1,n) by A0; :: thesis: verum