We must use a third party as witness. (as a witness)

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"There is a lot of poverty in the Third World." (in Third World countries)

To say "in the Third World." implies theres only one Third World. The way the sentence is structured seems incomplete.  You could also say "There is a lot of poverty in the Third World country known as Mozambique"

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Bonjour Gio !

Thanks very much for these remarks :)
In French, it's perfectly correct to use "le Tiers Monde" as a general expression for all under developed countries, but I agree it sounds weird in English.
I've now edited these examples accordingly.

Merci et à bientôt !