
theorem
  for S being non void Circuit-like non empty ManySortedSign for A
  being non-empty Circuit of S, s being State of A for x being set st x in
  InputVertices S holds s is_stable_at x
proof
  let S be non void Circuit-like non empty ManySortedSign;
  let A be non-empty Circuit of S;
  let s be State of A;
  let x be set;
  defpred P[Nat] means (Following(s,$1)).x = s.x;
  assume
A1: x in InputVertices S;
A2: now
    let n be Nat;
    assume
A3: P[n];
    (Following(s,n+1)).x = (Following Following(s,n)).x by Th12
      .= s.x by A1,A3,CIRCUIT2:def 5;
    hence P[n+1];
  end;
A4: P[0] by Th11;
  thus for n being Nat holds P[n] from NAT_1:sch 2(A4,A2);
end;
