Proposition 5.8. Assuming that:

• ${(A_j)}_{j\in J}$, $U$, $\sim$ as usual

• $n\in \mathbb{N}$

• for each $j$ we have $B_j,C_j\subseteq A_j^n$

• (1) $[{(B_j)}_{j\in J}]\cap [{(C_j)}_{j\in J}]=[{(B_j\cap C_j)}_{j\in J}]$
• (2) $[{(B_j)}_{j\in J}]\cup [{(C_j)}_{j\in J}]=[{(B_j\cup C_j)}_{j\in J}]$
• (3) $[{(B_j)}_{j\in J}]\setminus [{(C_j)}_{j\in J}]=[{(B_j\setminus C_j)}_{j\in J}]$
• (4) If $n>0$, $k\in \{1,\dots ,n\}$ then

  $${\pi }_k([{(B_j)}_{j\in J}])=[{({\pi }_k(B_j))}_{j\in J}].$$