The result Gini=2*AUROC-1 is hard to prove because it is not necessarily true. The Wikipedia article on the Receiver Operating Characteristic curve gives the result as a definition of Gini, and the article by Hand and Till (cited by nealmcb) merely says that the graphic definition of Gini using the ROC curve leads to this formula.
The catch is that this definition of Gini is used in the machine-learning and engineering communities, but a different definition is used by economists and demographers (going back to Gini's original paper). The Wikipedia article on the Gini coefficient sets out this definition, based on the Lorenz curve.
A paper by Schechtman & Schechtman (2016) sets out the relationship between AUC and the original Gini definition. But to see that they cannot be exactly the same, suppose that the proportion of events is p and that we have a perfect classifier. The ROC curve then passes through the top-left corner and AUCROC is 1. However, the (flipped) Lorenz curve runs from (0,0) to (p,1) to (1,1) and the economists' Gini is 1-p/2, which is nearly but not exactly 1.
If events are rare, then the relationship Gini=2*AUROC-1 is nearly but not exactly true using Gini's original definition. The relationship is only exactly true if Gini is redefined to make it true.