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 Th4:
  for B being non empty Subset of U0 st B=the carrier of U0 holds B
  is opers_closed & for o holds o/.B = o
proof
  let B be non empty Subset of U0;
  assume
A1: B=the carrier of U0;
A2: for o holds B is_closed_on o
  proof
    let o;
    let s be FinSequence of B;
    assume
A3: len s = arity o;
    dom o = (arity o)-tuples_on B & s is Element of (len s)-tuples_on B by A1,
FINSEQ_2:92,MARGREL1:22;
    then
A4: o.s in rng o by A3,FUNCT_1:def 3;
    rng o c= B by A1,RELAT_1:def 19;
    hence thesis by A4;
  end;
  for o holds o/.B = o
  proof
    let o;
    dom o = (arity(o))-tuples_on B & o/.B =o|((arity(o))-tuples_on B) by A1,A2
,Def5,MARGREL1:22;
    hence thesis by RELAT_1:68;
  end;
  hence thesis by A2;
end;
