Let $E$ be a vector space. Let $A,B,C$ be vector subspaces of $E$. Let $\oplus$ denotes the direct sum of vector spaces. Let $\cong$ denotes the vector space isomorphism. If $B \cong C$ then $A \oplus B \cong A \oplus C$. This property has been used in this thread. I would like to ask if the reverse holds, i.e.,
If $A \oplus B \cong A \oplus C$ then $B \cong C$.
Thank you so much for your elaboration!