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Yes, lets trust the people of Brazil, better yet lets trust the people of Brazil that actually live in the rain-forest, I wonder if they want to chop it all down?
Most and nearly 100% of the indigenous peoples do not. Certainly those in state of Para where the Brazil Nuts come from don't. BTW in Brazil they are called Para nuts. Para is a small state, entirely inside the Amazon.
Nor would those who illegally catch wild animals to sell, especially parrots. Most do live in some way using the forests and "ecotourism" especially via boats traveling up the rivers is increasing; however, the people cutting down valuable individual trees, and then burning the forest to hid their crime, live there too - that is how they make much better living than most. One good tree is worth more than a year's salary at the minimum wage.
The Amazon rain forest is huge - impossible to stop this illegal cutting, caused by the US and EU's demand for pretty wood furniture. The photo in your post is only a few years old - It show how big and impossible to control the Amazon rain forest -that small tribe was not even known to exist, prior to that photo, taking from an army helicopter on patrol.
If you want to stop the cutting and burning start something like PETA. I.e. instead of throwing blood on fur coats, make deep scratches with knifes in furniture of stores selling furniture made of Brazilian wood* and tell your friends with their pretty wood furniture, they helped finance the destruction of Brazilian rain forest.
It is pointless to blame Brazil - it is doing all it can to stop the illegal cutting and subsequent burning. Last year, as it cannot be stopped by forest police, Brazil has begun to license the extraction of valuable trees in small scattered areas so forest need not be burnt afterwards. Replanting more than cut is also part of the license requirement.
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* If quickly done by a dozen of so people with gloved hands to leave no finger prints on the quickly dropped knives, none can be found guilty as all can claim they were just in store shopping and did not have a knife, etc.