
theorem Th13:
  for S1,S2,S being non void Circuit-like non empty
ManySortedSign st InputVertices S1 misses InnerVertices S2 & 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,
s1 be State of A1 st s1 = s|the carrier of S1
for n being natural Number holds
Following(s, n)|the carrier of S1 = Following(s1, n)
proof
  let S1,S2,S be non void Circuit-like non empty ManySortedSign such that
A1: InputVertices S1 misses InnerVertices S2 & 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
A2: A1 tolerates A2 & A = A1+*A2;
  let s be State of A, s1 be State of A1 such that
A3: s1 = s|the carrier of S1;
  let n be natural Number;
A0: n is Nat by TARSKI:1;
  defpred P[Nat] means Following(s, $1)|the carrier of S1 = Following(s1, $1);
A4: now
    let n be Nat;
A5: Following(s, n+1) = Following Following(s, n) & Following Following(s1
    , n) = Following(s1, n+1) by FACIRC_1:12;
    assume P[n];
    hence P[n+1] by A1,A2,A5,Th10;
  end;
  Following(s, 0)|the carrier of S1 = s1 by A3,FACIRC_1:11
    .= Following(s1, 0) by FACIRC_1:11;
  then
A6: P[ 0 ];
  for n being Nat holds P[n] from NAT_1:sch 2(A6,A4);
  hence thesis by A0;
end;
