  Using the method of Coand\u{a} and Trautmann (2006), we give a simple proof
of the following theorem due to Tyurin (1976) in the smooth case: if a vector
bundle $E$ on a $c$-codimensional locally Cohen-Macaulay closed subscheme $X$
of the projective space $P^n$ extends to a vector bundle $F$ on a similar
closed subscheme $Y$ of $P^N$, for every $N > n$, then $E$ is the restriction
to $X$ of a direct sum of line bundles on $P^n$. Using the same method, we also
provide a proof of the Babylonian tower theorem for locally complete
intersection subschemes of projective spaces.
