
theorem Th22:
  for S1,S2,S being non void Circuit-like non empty
ManySortedSign st InputVertices S1 misses InnerVertices S2 & InputVertices S2
  misses InnerVertices S1 & S = S1+*S2 for A1 being non-empty Circuit of S1, A2
  being non-empty Circuit of S2 for A being non-empty Circuit of S st A1
tolerates A2 & A = A1+*A2 for n1,n2 being Nat, s being State of A for s1 being
State of A1 st s1 = s|the carrier of S1 for s2 being State of A2 st s2 = s|the
carrier of S2 & Following(s1, n1) is stable & Following(s2, n2) is stable holds
  Following(s, max(n1,n2)) is stable
proof
  let S1,S2,S be non void Circuit-like non empty ManySortedSign such that
A1: InputVertices S1 misses InnerVertices S2 and
A2: InputVertices S2 misses InnerVertices S1 and
A3: S = S1+*S2;
  let A1 be non-empty Circuit of S1, A2 be non-empty Circuit of S2;
  let A be non-empty Circuit of S such that
A4: A1 tolerates A2 and
A5: A = A1+*A2;
  let n1,n2 be Nat;
  let s be State of A;
  set n = max(n1,n2);
  let s0 be State of A1 such that
A6: s0 = s|the carrier of S1;
A7: Following(s, n)|the carrier of S1 = Following(s0, n) by A1,A3,A4,A5,A6,Th13
;
  S1 tolerates S2 by A4,CIRCCOMB:def 3;
  then
A8: S1+*S2 = S2+*S1 by CIRCCOMB:5;
  let s3 be State of A2 such that
A9: s3 = s|the carrier of S2 and
A10: Following(s0, n1) is stable and
A11: Following(s3, n2) is stable;
  A1+*A2 = A2+*A1 by A4,CIRCCOMB:22;
  then
A12: Following(s, n)|the carrier of S2 = Following(s3, n) by A2,A3,A4,A5,A9,A8
,Th13,CIRCCOMB:19;
A13: Following(s3, n) is stable by A11,Th4,XXREAL_0:25;
A14: Following(s0, n) is stable by A10,Th4,XXREAL_0:25;
  thus Following(s, max(n1,n2)) = Following(s0, n)+*Following(s3, n) by A1,A2
,A3,A4,A5,A6,A9,Th21
    .= (Following Following(s0, n))+*Following(s3, n) by A14
    .= (Following Following(s0, n))+*Following Following(s3, n) by A13
    .= Following Following(s, n) by A2,A3,A4,A5,A7,A12,CIRCCOMB:32;
end;
