let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InputVertices S1 misses InnerVertices S2 & 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 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,(max (n1,n2))) is stable )

assume A1: ( InputVertices S1 misses InnerVertices S2 & 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 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,(max (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,(max (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,(max (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,(max (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,(max (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,(max (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,(max (n1,n2))) is stable

let s be State of A; :: thesis: Following (s,(max (n1,n2))) is stable

A5: the Sorts of A1 tolerates the Sorts of A2 by A2, CIRCCOMB:def 3;

then reconsider s0 = s | the carrier of S1 as State of A1 by A3, CIRCCOMB:26;

reconsider s3 = s | the carrier of S2 as State of A2 by A3, A5, CIRCCOMB:26;

( Following (s0,n1) is stable & Following (s3,n2) is stable ) by A4;

hence Following (s,(max (n1,n2))) is stable by A1, A2, A3, Th22; :: 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 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,(max (n1,n2))) is stable )

assume A1: ( InputVertices S1 misses InnerVertices S2 & 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 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,(max (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,(max (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,(max (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,(max (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,(max (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,(max (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,(max (n1,n2))) is stable

let s be State of A; :: thesis: Following (s,(max (n1,n2))) is stable

A5: the Sorts of A1 tolerates the Sorts of A2 by A2, CIRCCOMB:def 3;

then reconsider s0 = s | the carrier of S1 as State of A1 by A3, CIRCCOMB:26;

reconsider s3 = s | the carrier of S2 as State of A2 by A3, A5, CIRCCOMB:26;

( Following (s0,n1) is stable & Following (s3,n2) is stable ) by A4;

hence Following (s,(max (n1,n2))) is stable by A1, A2, A3, Th22; :: thesis: verum