
theorem Th2:
  for S being non empty non void ManySortedSign for A being
MSAlgebra over S for o being OperSymbol of S, a being Function st a in Args(o,A
) holds dom a = dom the_arity_of o & for i being set st i in dom the_arity_of o
  holds a.i in (the Sorts of A).((the_arity_of o)/.i)
proof
  let S be non empty non void ManySortedSign;
  let A be MSAlgebra over S;
  let o be OperSymbol of S;
  let x be Function;
  assume x in Args(o,A);
  then x in product((the Sorts of A) * (the_arity_of o)) by PRALG_2:3;
  then
A1: ex f being Function st x = f & dom f = dom ((the Sorts of A)*
  the_arity_of o) &
for y be object st y in dom ((the Sorts of A)*the_arity_of o)
  holds f.y in ((the Sorts of A)*the_arity_of o).y by CARD_3:def 5;
  hence
A2: dom x = dom the_arity_of o by PARTFUN1:def 2;
  let y be set;
  assume
A3: y in dom the_arity_of o;
  then
A4: (the_arity_of o).y = (the_arity_of o)/.y by PARTFUN1:def 6;
  x.y in ((the Sorts of A)*the_arity_of o).y by A1,A2,A3;
  hence thesis by A3,A4,FUNCT_1:13;
end;
