
theorem Th35:
  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 holds (Following s).[p,f] = f.(s*p)
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: g`1 = p & g`2 = f;
A2: the_result_sort_of g = (the ResultSort of S).g by MSUALG_1:def 2
    .= g by CIRCCOMB:44;
  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;
  hence thesis by A1,A2,Th34;
end;
