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I believe that your conjecture "forecast of a sum it should be equal the sum of a forecasts:" can only be true if the individual forecasts can be shown to be completely independent. In "Optimal combination forecasts for hierarchical time series" by Rob J.Hyndman, Roman A.Ahmed, et al., reproduced in full here, the authors discuss this topic in detail. The key point they make here is the following
The various components of the hierarchy can interact in varying and complex ways. A change in one series at one level, can have a consequential impact on other series at the same level, as well as series at higher and lower levels
Of course, it is also possible to forecast all series at all levels independently, but this has the undesirable consequence of the higher level forecasts not being equal to the sum of the lower level forecasts. Consequently, if this method is used, some adjustment is then carried out to ensure the forecasts add up appropriately. These adjustments are usually done in an ad hoc manner.
They then proceed to discuss a method whereby they use a regression model to reconcile and combine the forecasts across all the levels.