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Is this really true? And does is explain weight-loss due to calorie restriction (based on the explanation it seems like the weight loss should take a lot of time without exercise -- need to breath out all that CO2).
The effect of reducing calorie intake is so much greater than exercising more. If you could reduce your calories by 500/day that would be like working out vigorously for 1 hour every day. Its all just inputs and outputs. If your baseline calorie burn assuming no exercise is 1500/day and you reduce from 1750 to 1250, then over time you will lose weight.

In either case, the mechanism is the same, you convert organic carbon compounds to CO2 and water and excreting those through breathing and urination is how the weight goes from your body back to the environment.

It's the reverse mechanism of how potted plants grow without the soil being depleted, the mass of the growing plant comes from the H2O and CO2 in the atmosphere to perform photosynthesis, among other reactions.