reserve I for non empty set,
  J for ManySortedSet of I,
  S for non void non empty ManySortedSign,
  i for Element of I,
  c for set,
  A for MSAlgebra-Family of I,S,
  EqR for Equivalence_Relation of I,
  U0,U1,U2 for MSAlgebra over S,
  s for SortSymbol of S,
  o for OperSymbol of S,
  f for Function;

theorem Th16:
  for i be Element of I for n be set st n in dom(the_arity_of o)
for s be SortSymbol of S st s = (the_arity_of o).n for y be Element of Args(o,
  product A) for g be Function st g = y.n holds g.i in (the Sorts of A.i).s
proof
  let i be Element of I;
  let n be set;
  assume n in dom(the_arity_of o);
  then
A1: n in dom ((the Sorts of (product A))*(the_arity_of o)) by PRALG_2:3;
  let s be SortSymbol of S such that
A2: s = (the_arity_of o).n;
  let y be Element of Args(o,product A);
  y in Args(o,product A);
  then
A3: y in product ((the Sorts of (product A))*(the_arity_of o)) by PRALG_2:3;
  let g be Function;
  assume g = y.n;
  then g in ((the Sorts of (product A))*(the_arity_of o)).n by A1,A3,CARD_3:9;
  then g in (the Sorts of (product A)).s by A2,A1,FUNCT_1:12;
  then
A4: g in product Carrier(A,s) by PRALG_2:def 10;
  i in I;
  then
A5: i in dom (Carrier(A,s)) by PARTFUN1:def 2;
  ex U0 being MSAlgebra over S st U0 = A.i & (Carrier(A,s)) .i = (the
  Sorts of U0).s by PRALG_2:def 9;
  hence thesis by A5,A4,CARD_3:9;
end;
