
theorem Th13:
  for S being Circuit-like non void non empty ManySortedSign for
  A being non-empty Circuit of S, s being State of A for n,m being Nat holds
  Following(s,n+m) = Following(Following(s,n),m)
proof
  let S be Circuit-like non void non empty ManySortedSign;
  let A be non-empty Circuit of S, s be State of A;
  let n be Nat;
  defpred P[Nat] means Following(s,n+$1) = Following(Following(s,n),$1);
A1: for m being Nat st P[m] holds P[m+1]
  proof
    let m be Nat;
    assume
A2: Following(s,n+m) = Following(Following(s,n),m);
    thus Following(s,n+(m+1)) = Following(s,n+m+1)
      .= Following Following(s,n+m) by Th12
      .= Following(Following(s,n),m+1) by A2,Th12;
  end;
A3: P[0] by Th11;
  thus for i being Nat holds P[i] from NAT_1:sch 2(A3,A1);
end;
