Mathematica's Interval Arithmetic is capable of handling the uncertainty in a rational manner

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Interval Arithmetic with Symbolic Intervals

I've run into a problem with IntervalUnion and IntervalIntersection when dealing with symbolic intervals. The following code ...
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How to code a For-loop within Interval correctly?

Help me please to write a For-loop correctly. I've implemented the 1-st step of an algorithm and finally it outputs two intervals: ...