The existence of a non-hereditarily complete family in an arbitrary separable Banach space

It is proved that every separable Banach space $E$ contains a complete minimal family $\{x_j\}_1^\infty$ with the total biorthogonal family $\{f_j\}_1^\infty$ (in $E^*$) but not hereditarily complete (this means that the closed linear envelope of the f amily $\{f_j(z)x_j\}_1^\infty)$ does not coincide with $E$).