  In this note we observe that in the hyper-graph removal lemma the edge
removal can be done in a way that the symmetries of the original hyper-graph
remain preserved. As an application we prove the following generalization of
Szemer\'edi's Theorem on arithmetic progressions. If in an Abelian group $A$
there are sets $S_1,S_2...,S_t$ such that the number of arithmetic progressions
$x_1,x_2,...,x_t$ with $x_i\in S_i$ is $o(|A|^2)$ then we can shrink each $S_i$
by $o(|A|)$ elements such that the new sets don't have such a diagonal
arithmetic progression.
