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 n1, n2 being Nat st ( for s being State of A1 holds Following (s,n1) is stable ) & ( for s being State of A2 holds Following (s,n2) is stable ) holds
for s being State of A holds Following (s,(n1 + n2)) is stable )
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 n1, n2 being Nat st ( for s being State of A1 holds Following (s,n1) is stable ) & ( for s being State of A2 holds Following (s,n2) is stable ) holds
for s being State of A holds Following (s,(n1 + n2)) is stable
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 n1, n2 being Nat st ( for s being State of A1 holds Following (s,n1) is stable ) & ( for s being State of A2 holds Following (s,n2) is stable ) holds
for s being State of A holds Following (s,(n1 + n2)) is stable
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 n1, n2 being Nat st ( for s being State of A1 holds Following (s,n1) is stable ) & ( for s being State of A2 holds Following (s,n2) is stable ) holds
for s being State of A holds Following (s,(n1 + n2)) is stable
let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for n1, n2 being Nat st ( for s being State of A1 holds Following (s,n1) is stable ) & ( for s being State of A2 holds Following (s,n2) is stable ) holds
for s being State of A holds Following (s,(n1 + n2)) is stable )
assume that
A2: A1 tolerates A2 and
A3: A = A1 +* A2 ; :: thesis: for n1, n2 being Nat st ( for s being State of A1 holds Following (s,n1) is stable ) & ( for s being State of A2 holds Following (s,n2) is stable ) holds
for s being State of A holds Following (s,(n1 + n2)) is stable
let n1, n2 be Nat; :: thesis: ( ( for s being State of A1 holds Following (s,n1) is stable ) & ( for s being State of A2 holds Following (s,n2) is stable ) implies for s being State of A holds Following (s,(n1 + n2)) is stable )
assume A4: ( ( for s being State of A1 holds Following (s,n1) is stable ) & ( for s being State of A2 holds Following (s,n2) is stable ) ) ; :: thesis: for s being State of A holds Following (s,(n1 + n2)) is stable
let s be State of A; :: thesis: Following (s,(n1 + n2)) is stable
A5: the Sorts of A1 tolerates the Sorts of A2 by A2, CIRCCOMB:def 3;
then reconsider s1 = s | the carrier of S1 as State of A1 by A3, CIRCCOMB:26;
reconsider s2 = (Following (s,n1)) | the carrier of S2 as State of A2 by A3, A5, CIRCCOMB:26;
( Following (s1,n1) is stable & Following (s2,n2) is stable ) by A4;
hence Following (s,(n1 + n2)) is stable by A1, A2, A3, Th19; :: thesis: verum