reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;

theorem Th22:
  F is commutative & F is associative & F is idempotent implies
  for a being Element of Y st f.:B = {a} holds F$$(B,f) = a
proof
  assume
A1: F is commutative & F is associative & F is idempotent;
  let a be Element of Y;
  assume
A2: f.:B = {a};
A3: for b being Element of X st b in B holds f.b = a
  proof
    let b be Element of X;
    assume b in B;
    then f.b in {a} by A2,FUNCT_2:35;
    hence thesis by TARSKI:def 1;
  end;
  B <> {} by A2;
  hence thesis by A1,A3,Th21;
end;
