let S1, S2, S be non empty non void Circuit-like ManySortedSign ; ( 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 )
; 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; 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; 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; ( 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
; 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; ( ( 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 ) )
; for s being State of A holds Following (s,(max (n1,n2))) is stable
let s be State of A; 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; verum