Below let's work over coherent sheaves on a smooth projective algebraic curve. 


We call a subsheaf $\mathcal{F'}$ of $\mathcal{F}$ saturated if it $\mathcal{F/F'}$ is locally free.


We call a locally free sheaf indecomposible if it cannot be written as a direct sum of two saturated subsheaves.


Suppose $\mathcal{E=F\_1 \oplus F\_2\oplus\dots \oplus F\_n=G\_1 \oplus G\_2\oplus\dots \oplus G\_m}$ where the summands are locally free and indecomposible in the sense above.


Then how can we prove $m=n$ and the summands are isomorphic after a permutation?


Thanks for comments and references!



Unique decomposition of locally free sheaf
This is proven by Atiyah (Theorem 3) in [On the Krull-Schmidt theorem with application to sheaves](http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1956__84_/BSMF_1956__84__307_0/BSMF_1956__84__307_0.pdf). I think it's worth noting that techniques used are very broadly applicable; he really just uses that the algebra $A=\mathrm{End}(\mathcal{E})$ is artinian (since it is finite dimensional over a field). 


**EDIT:** As Sasha notes below, the saturated hypothesis is a red herring. Any Remak decomposition of $\mathcal{E}$ will automatically be into saturated subsheaves.


In a more modern perspective, I think we would just note that the Remak decompositions above correspond canonically to Remak decompositions $A\cong \mathrm{Hom}(\mathcal{E},\mathcal{F}\_1)\oplus \cdots\oplus\mathrm{Hom}(\mathcal{E},\mathcal{F}\_n)$, etc. of $A$ as a left module over itself: such a Remak decomposition must be of the form $A\cong Ae\_1\oplus \cdots \oplus Ae\_n$ for idempotents $e\_n$, and $\mathcal{F}\_i$ is the image of $e\_i$. We can then use the usual Krull-Schmidt theorem from artinian rings.


