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 A1: ( InputVertices S2 misses InnerVertices S1 & 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 A2: ( A1 tolerates A2 & 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 A2, CIRCCOMB:def 3;
then ( S = S2 +* S1 & A2 tolerates A1 & A = A2 +* A1 ) by A1, A2, CIRCCOMB:9, CIRCCOMB:23, CIRCCOMB:26;
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, Th14; :: thesis: verum