Generalised acyclic edge colourings of graphs with large girth

The r-acyclic edge chromatic number of a graph G is the minimum number of colours required to colour the edges of G in such a way that adjacent edges receive different colours and every cycle C receives at least image colours. We prove that for any integer image, the r-acyclic edge chromatic number of any graph G with maximum degree image and with girth at least image is at most image.