1. Is it true that if $A_1\times A_2\times ... \times A_n = B_1\times B_2\times .. \times B_m$, where $A_i, B_j$ are homotopy types of connected complexes not decomposable into a product, then the multisets $A_i$ and $B_j$ coincide? What if it is limited only to finite complexes?

 2. Is it true that if $A_1​​\wedge A_2 \wedge .. \wedge A_n = B_1\wedge B_2 \wedge .. \wedge B_m$, where $A_i, B_j$ are homotopy types of connected pointed complexes not decomposable into a smash product, then the multisets $A_i$ and $B_j$ coincide? What if it is limited only to finite complexes?

It is clear that finite complexes decompose into a finite number of indecomposable (with smash product, the connectedness increases, and with a Cartesian product, the homology groups increase). 

**Update**: in fact, the point is decomposable with respect to the smash of the product, so there are no non-decomposable complexes with respect to the smash of the product. With respect to the direct product from the homotopy groups of the product, the indecomposability of the point is obvious, so my proof remains valid with respect to it.

P.S. [Similar question](https://mathoverflow.net/questions/420162/is-the-decomposition-of-the-homotopy-type-of-a-complex-into-a-bouquet-unique/420325#420325)