
theorem Th10:
  for S being Circuit-like non void non empty ManySortedSign for
A being non-empty Circuit of S for s being State of A, g being Gate of S holds
  (Following s).the_result_sort_of g = Den(g, A).(s*the_arity_of g)
proof
  let S be Circuit-like non void non empty ManySortedSign;
  let A be non-empty Circuit of S;
  let s be State of A, g be Gate of S;
  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
A1: 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 A1,CIRCUIT2:def 5;
end;
