reserve IIG for Circuit-like non void non empty ManySortedSign;
reserve IIG for monotonic Circuit-like non void non empty ManySortedSign;

theorem
  for IIG being non void Circuit-like non empty ManySortedSign for SCS
being non-empty Circuit of IIG, s being State of SCS, o being OperSymbol of IIG
  holds (Den(o,SCS)).(o depends_on_in s) in (the Sorts of SCS).(
  the_result_sort_of o)
proof
  let IIG be non void Circuit-like non empty ManySortedSign;
  let SCS be non-empty Circuit of IIG, s be State of SCS, o be OperSymbol of
  IIG;
A1: dom the ResultSort of IIG = the carrier' of IIG by FUNCT_2:def 1;
  Result(o,SCS) = ((the Sorts of SCS) * the ResultSort of IIG).o by
MSUALG_1:def 5
    .= (the Sorts of SCS).((the ResultSort of IIG).o) by A1,FUNCT_1:13
    .= (the Sorts of SCS).(the_result_sort_of o) by MSUALG_1:def 2;
  hence thesis by FUNCT_2:5;
end;
