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We give a proof of local strong factorization of a birational, monomial, extension of regular local rings along a valuation of rank 1 and maximal rational rank. Our proof uses methods from linear algebra, and is in the spirit of Christensen's proof of this result in dimension 3. This has also been proven by Karu using toric geometry.doi:10.1215/ijm/1258735323 fatcat:iynqa3rfmbc53avblxfbdemsnu