>>7839570aí: close to the one you are speaking to
A: onde está meu lenço?
B: aí do teu lado.
(its different from aqui, then the subject you are refering to is close to the one who speaks)
Ali: in a place nearby/at the range of view
A: onde está meu lenço?
B: ali em cima da mesa.
lá: somewhere not in the range of view or not close to both comunicator and receiever or in range of view but far away.
1:
A: Onde está meu lenço?
B: deixaste lá perto do quarto
2:
A: Onde está meu lenço?
B: Lá do outro lado do rio
(note that in situation 2 Ali can also be used since the other side of the river can be seen by both comunicator and receiever)
Haver: a verb what can be flexionated in many different forms, meaning in a general way something what exist, existed or may exist somwhere close or far from the communicator or receiever. the meanings of "haver" are relative to the way it is conjugated and the uses of aí, alí, alá, acolá, aqui, acá, ante, depois and entre before or after the verb haver or its forms.
Conjugation of verb "haver":
https://www.conjugacao.com.br/verbo-haver/ 
