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 s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,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 s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,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 s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,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 s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n)

let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n) )

assume that

A2: A1 tolerates A2 and

A3: A = A1 +* A2 ; :: thesis: for s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n)

let s1 be State of A1; :: thesis: for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n)

let s2 be State of A2; :: thesis: for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n)

let s be State of A; :: thesis: ( s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable implies for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n) )

assume that

A4: s1 = s | the carrier of S1 and

A5: s2 = s | the carrier of S2 and

A6: s1 is stable ; :: thesis: for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n)

defpred S_{3}[ Nat] means (Following (s,$1)) | the carrier of S2 = Following (s2,$1);

.= Following (s2,0) by FACIRC_1:11 ;

then A10: S_{3}[ 0 ]
;

thus for n being Nat holds S_{3}[n]
from NAT_1:sch 2(A10, A7); :: 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 s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,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 s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,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 s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,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 s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n)

let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n) )

assume that

A2: A1 tolerates A2 and

A3: A = A1 +* A2 ; :: thesis: for s1 being State of A1

for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n)

let s1 be State of A1; :: thesis: for s2 being State of A2

for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n)

let s2 be State of A2; :: thesis: for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds

for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n)

let s be State of A; :: thesis: ( s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable implies for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n) )

assume that

A4: s1 = s | the carrier of S1 and

A5: s2 = s | the carrier of S2 and

A6: s1 is stable ; :: thesis: for n being Nat holds (Following (s,n)) | the carrier of S2 = Following (s2,n)

defpred S

A7: now :: thesis: for n being Nat st S_{3}[n] holds

S_{3}[n + 1]

(Following (s,0)) | the carrier of S2 =
s2
by A5, FACIRC_1:11
S

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

A8: ( Following (s,(n + 1)) = Following (Following (s,n)) & Following (Following (s2,n)) = Following (s2,(n + 1)) ) by FACIRC_1:12;

the Sorts of A1 tolerates the Sorts of A2 by A2, CIRCCOMB:def 3;

then reconsider Fs1 = (Following (s,n)) | the carrier of S1 as State of A1 by A3, CIRCCOMB:26;

Following (s1,n) = Fs1 by A1, A2, A3, A4, Th13;

then A9: Fs1 is stable by A6, Th3;

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

hence S_{3}[n + 1]
by A1, A2, A3, A8, A9, Th15; :: thesis: verum

end;A8: ( Following (s,(n + 1)) = Following (Following (s,n)) & Following (Following (s2,n)) = Following (s2,(n + 1)) ) by FACIRC_1:12;

the Sorts of A1 tolerates the Sorts of A2 by A2, CIRCCOMB:def 3;

then reconsider Fs1 = (Following (s,n)) | the carrier of S1 as State of A1 by A3, CIRCCOMB:26;

Following (s1,n) = Fs1 by A1, A2, A3, A4, Th13;

then A9: Fs1 is stable by A6, Th3;

assume S

hence S

.= Following (s2,0) by FACIRC_1:11 ;

then A10: S

thus for n being Nat holds S