
theorem Th34:
  for S being gate`2=den Circuit-like non void non empty
  ManySortedSign for A being non-empty Circuit of S st A is gate`2=den for s
being State of A, g being Gate of S holds (Following s).the_result_sort_of g =
  g`2.(s*the_arity_of g)
proof
  let S be gate`2=den Circuit-like non void non empty ManySortedSign;
  let A be non-empty Circuit of S such that
A1: for g being set st g in the carrier' of S holds g = [g`1, (the
  Charact of A).g];
  let s be State of A, g be Gate of S;
A2: Den(g, A) = (the Charact of A).g by MSUALG_1:def 6
    .= [g`1, (the Charact of A).g]`2
    .= g`2 by A1;
  set v = the_result_sort_of g;
  dom the ResultSort of S = the carrier' of S by FUNCT_2:def 1;
  then (the ResultSort of S).g in rng the ResultSort of S by FUNCT_1:def 3;
  then
A3: v in InnerVertices S by MSUALG_1:def 2;
  then g depends_on_in s = s*the_arity_of g & action_at v = g by CIRCUIT1:def 3
,MSAFREE2:def 7;
  hence thesis by A2,A3,CIRCUIT2:def 5;
end;
