reserve S for non void non empty ManySortedSign,
  U1,U2 for MSAlgebra over S,
  o for OperSymbol of S,
  n for Nat;

theorem Th6:
  for U1 being MSAlgebra over S
  for x be Function st x in Args(o,U1) holds
  dom x = dom the_arity_of o &
  for y be set st y in dom ((the Sorts of U1) * (the_arity_of o)) holds
    x.y in ((the Sorts of U1) * (the_arity_of o)).y
proof
  let U1 be MSAlgebra over S;
  let x be Function;
A1: Args(o,U1) = product((the Sorts of U1) * (the_arity_of o)) by PRALG_2:3;
  dom (the Sorts of U1) = (the carrier of S) by PARTFUN1:def 2;
  then
A2: rng (the_arity_of o) c= dom (the Sorts of U1) by FINSEQ_1:def 4;
  assume
A3: x in Args(o,U1);
  then dom x = dom ((the Sorts of U1) * (the_arity_of o)) by A1,CARD_3:9;
  hence thesis by A3,A1,A2,CARD_3:9,RELAT_1:27;
end;
