reserve U0 for Universal_Algebra,
  U1 for SubAlgebra of U0,
  o for operation of U0;

theorem
  for H being non empty Subset of U0 for o holds arity o = 0 implies (H
  is_closed_on o iff o.{} in H)
proof
  let H be non empty Subset of U0;
  let o;
  assume
A1: arity o = 0;
  thus H is_closed_on o implies o.{} in H
  proof
    assume
A2: H is_closed_on o;
    consider s being FinSequence of H such that
A3: len s = arity o by FINSEQ_1:19;
    s = {} by A1,A3;
    hence thesis by A2,A3;
  end;
  thus o.{} in H implies H is_closed_on o
  proof
    assume
A4: o.{} in H;
    let s be FinSequence of H;
    assume len s = arity o;
    then s = {} by A1;
    hence thesis by A4;
  end;
end;
