theorem Th52:
  a is_a_unity_wrt o implies {a} is_a_unity_wrt o.:^2 & o.:^2 is
  having_a_unity & the_unity_wrt o.:^2 = {a}
proof
  assume
A1: a is_a_unity_wrt o;
  now
    let x be Subset of D;
    thus (o.:^2).({a},x) = o.:[:{a},x:] by Th44
      .= D /\ x by A1,Th51
      .= x by XBOOLE_1:28;
    thus (o.:^2).(x,{a}) = o.:[:x,{a}:] by Th44
      .= D /\ x by A1,Th51
      .= x by XBOOLE_1:28;
  end;
  hence
A2: {a} is_a_unity_wrt o.:^2 by BINOP_1:3;
  hence ex A being Subset of D st A is_a_unity_wrt o.:^2;
  thus thesis by A2,BINOP_1:def 8;
end;
