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

theorem
  Constants(U0) = { o.{} where o is operation of U0: arity o = 0 }
proof
  set S = { o.{} where o is operation of U0: arity o = 0 };
  thus Constants(U0) c= S
  proof
    let a be object;
    assume a in Constants(U0);
    then consider u being Element of U0 such that
A1: u = a and
A2: ex o be operation of U0 st arity o = 0 & u in rng o;
    consider o be operation of U0 such that
A3: arity o = 0 and
A4: u in rng o by A2;
    consider a2 being object such that
A5: a2 in dom o and
A6: u = o.a2 by A4,FUNCT_1:def 3;
    dom o = 0 -tuples_on the carrier of U0 by A3,MARGREL1:22;
    then a2 is Tuple of 0,the carrier of U0 by A5,FINSEQ_2:131;
    then reconsider a1 = a2 as FinSequence of the carrier of U0;
    len a1 = 0 by A3,A5,MARGREL1:def 25;
    then a1 = {};
    hence thesis by A1,A3,A6;
  end;
  thus S c= Constants(U0)
  proof
    let a be object;
    assume a in S;
    then consider o being operation of U0 such that
A7: a = o.{} and
A8: arity o = 0;
    dom o = 0-tuples_on the carrier of U0 by A8,MARGREL1:22
      .={<*>the carrier of U0} by FINSEQ_2:94;
    then {}the carrier of U0 in dom o by TARSKI:def 1;
    then o.({}the carrier of U0) in rng o by FUNCT_1:def 3;
    hence thesis by A7,A8;
  end;
end;
