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Minimum Spanning Tree

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Given a graph G and a minimum spanning tree T, suppose that we decrease the weight of one of the edges in T. Show that T is still a minimum spanning tree for G. More formally, let T be a minimum spanning tree for G with edge weights given by weight function w. Choose one edge (x, y) is a subset of T and a positive number k, and define the weight function w' by

w'(u,v) = w(u,v) if (u, v) =/ (x,y)
w(x,y) - k if (u,v) = (x,y)

Show that T is a minimum spanning tree for G with edge weights given by w'.

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