
theorem
  for n being Nat, X being finite non empty set, f being Function of n
-tuples_on X, X, p being FinSeqLen of n for s being State of 1GateCircuit(p,f)
  holds (Following s).[p,f] = f.(s*p)
proof
  let n be Nat;
  let X be non empty finite set;
  let f be Function of n-tuples_on X, X;
  let p be FinSeqLen of n;
  let s be State of 1GateCircuit(p,f);
  set S = 1GateCircStr(p,f), A = 1GateCircuit(p,f);
  set IV = InnerVertices S;
  IV = {[p,f]} by Th42;
  then reconsider v = [p,f] as Element of IV by TARSKI:def 1;
  the carrier' of S = {[p,f]} by Def6;
  then reconsider o = [p,f] as Gate of S by TARSKI:def 1;
  the_result_sort_of o = v by Def6;
  then
A1: action_at v = o by MSAFREE2:def 7;
  the_arity_of o = p by Def6;
  then
A2: o depends_on_in s = s*p by CIRCUIT1:def 3;
  (Following s).v = (Den (action_at v, A)).(action_at v depends_on_in s)
  by CIRCUIT2:def 5;
  hence thesis by A1,A2,Def13;
end;
