let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InputVertices S2 misses InnerVertices S1 & 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 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 that

A1: InputVertices S2 misses InnerVertices S1 and

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

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 that

A3: A1 tolerates A2 and

A4: A = A1 +* A2 ; :: thesis: for s being State of A

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)

S1 tolerates S2 by A3, CIRCCOMB:def 3;

then A5: S = S2 +* S1 by A2, CIRCCOMB:5;

A = A2 +* A1 by A3, A4, CIRCCOMB:22;

hence for s being State of A

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) by A1, A3, A5, Th13, CIRCCOMB:19; :: 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 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 that

A1: InputVertices S2 misses InnerVertices S1 and

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

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 that

A3: A1 tolerates A2 and

A4: A = A1 +* A2 ; :: thesis: for s being State of A

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)

S1 tolerates S2 by A3, CIRCCOMB:def 3;

then A5: S = S2 +* S1 by A2, CIRCCOMB:5;

A = A2 +* A1 by A3, A4, CIRCCOMB:22;

hence for s being State of A

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) by A1, A3, A5, Th13, CIRCCOMB:19; :: thesis: verum