reserve A for set;
reserve X,Y,Z for set,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve u for UnOp of A,
  o,o9 for BinOp of A,
  a,b,c,e,e1,e2 for Element of A;

theorem Th3:
  e is_a_unity_wrt o iff for a holds o.(e,a) = a & o.(a,e) = a
proof
  thus e is_a_unity_wrt o implies for a holds o.(e,a) = a & o.(a,e) = a
  proof
    assume e is_a_left_unity_wrt o & e is_a_right_unity_wrt o;
    hence thesis;
  end;
  assume for a holds o.(e,a) = a & o.(a,e) = a;
  hence (for a holds o.(e,a) = a) & for a holds o.(a,e) = a;
end;
