Intervals and Distances: Question about a solution

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The text of a puzzle could be found in [1]. I have a question concerning the solution of Peter Winkler (that is the same as in the origin in Russian). He claims that if a and b are two (disjoint) intervals and I is interval of distances obtainable by taking one point from a and one from b then I has length equal to the sum of lengths of a and b. But it seems to be wrong because it does not depend from positions of the a and b. For example, if the left bound of a is 0 and the right bound of b is 1 that distance between these points is 1. So I need explanation of the proof of Winkler or may be some other proof. [1]: Peter Winkler's Mathematical Puzzle "Intervals and Distances"