
theorem
  for S being gate`1=arity gate`2isBoolean unsplit non void non empty
  ManySortedSign for A being Boolean gate`2=den non-empty Circuit of S for s
  being State of A, p being FinSequence, f being Function st [p,f] in the
  carrier' of S & for x being set st x in rng p holds s is_stable_at x holds
  Following s is_stable_at [p,f]
proof
  let S be gate`1=arity gate`2isBoolean unsplit non void non empty
  ManySortedSign;
  let A be Boolean gate`2=den non-empty Circuit of S;
  let s be State of A, p be FinSequence, f be Function;
  assume [p,f] in the carrier' of S;
  then reconsider g = [p,f] as Gate of S;
A1: the_arity_of g = (the Arity of S).g by MSUALG_1:def 1
    .= [(the Arity of S).g, g`2]`1
    .= g`1 by CIRCCOMB:def 8
    .= p;
A2: the_result_sort_of g = (the ResultSort of S).g by MSUALG_1:def 2
    .= g by CIRCCOMB:44;
  assume for x being set st x in rng p holds s is_stable_at x;
  hence thesis by A1,A2,Th19;
end;
