
theorem Th21:
  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 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
for n being natural Number holds
Following(s, n) = Following(s1, n)+*Following(s2, n)
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 s be State of A;
  let s1 be State of A1 such that
A6: s1 = s|the carrier of S1;
  let s2 be State of A2 such that
A7: s2 = s|the carrier of S2;
  let n be natural Number;
A8: Following(s, n)|the carrier of S1 = Following(s1, n) by A1,A3,A4,A5,A6,Th13
;
A9: dom Following(s, n) = the carrier of S & the carrier of S = (the
  carrier of S1) \/ the carrier of S2 by A3,CIRCCOMB:def 2,CIRCUIT1:3;
  S1 tolerates S2 by A4,CIRCCOMB:def 3;
  then
A10: S1+*S2 = S2+*S1 by CIRCCOMB:5;
  A1+*A2 = A2+*A1 by A4,CIRCCOMB:22;
  then Following(s, n)|the carrier of S2 = Following(s2, n) by A2,A3,A4,A5,A7
,A10,Th13,CIRCCOMB:19;
  hence thesis by A8,A9,FUNCT_4:70;
end;
