reserve U0,U1,U2,U3 for Universal_Algebra,
  n for Nat,
  x,y for set;
reserve A for non empty Subset of U0,
  o for operation of U0,
  x1,y1 for FinSequence of A;

theorem
  for U1 be Universal_Algebra, A be non empty Subset of U1, o be
  operation of U1 st A is_closed_on o holds arity (o/.A) = arity o
proof
  let U1 be Universal_Algebra, A be non empty Subset of U1, o be operation of
  U1;
  assume A is_closed_on o;
  then o/.A =o|((arity o)-tuples_on A) by Def5;
  then dom (o/.A) = dom o /\ ((arity o)-tuples_on A) by RELAT_1:61;
  then
A1: dom (o/.A) = ((arity o)-tuples_on the carrier of U1) /\ ((arity o)
  -tuples_on A) by MARGREL1:22
    .= (arity o)-tuples_on A by MARGREL1:21;
  dom (o/.A)=(arity (o/.A))-tuples_on A by MARGREL1:22;
  hence thesis by A1,FINSEQ_2:110;
end;
