Y$ be the embedding map, let $T:Y\to G$ be a bounded linear operator and assume that $S=TJ$. Furthermore, let $\nu_1,\nu_2\in {{\mathbb N}}$ be any numbers with $$\label{KB1}
e^{-\nu_1/8}+e^{-\nu_2/8}-e^{-(\nu_1+\nu_2)/8}\le 1/4.$$ Along with $\mathcal{P}=(F,G,S,{{\mathbb K}},\Lambda)$ we consider the problems $(F,Y,J,{{\mathbb K}},\Lambda)$ and $(B_Y,G,T,{{\mathbb K}},\Lambda)$. Then for all $n_1, n_2\in {{\mathbb N}}_0$, $$\label{AJ4}
e^{{\rm q }}_{\nu_1 n_1+2\nu_2 n_2}(S,F)
\le 
4 e^{{\rm q }}_{n_1}(J,F)\,e^{{\rm q }}_{n_2}(T,B_Y).$$ If, moreover, $F$ is a precompact subset of $Y$ and $\Lambda$ is linearly independent over $F$, then $$\label{QT3}
e^{{\rm q }}_{\nu_1 n_1+2\nu_2 n_2}(S,F)
\le 
4e^{{\rm q }}_{n_1}(J,F)\sup_{\mathcal{E}\in\mathscr{C}(B_Y)} e^{{\rm q }}_{n_2}(T, \mathcal{E}).$$

The first part, relation (\[AJ4\]), was proved in [@Hei04a], Proposition 1 and Corollary 3. As already mentioned at the end of the previous section, this result was formulated for a slightly less general type of numerical problem, but the proof of (\[AJ4\]) is literally the same as that of Proposition 1 and Corollary 3 in [@Hei04a].

The specific form of multiplicativity stated in (\[QT3\]) (a single $e_n^{{\rm q }}(S,F)$ is replaced by a supremum over a family of subsets of $F$) will be needed in sections 5 and 6. For further explanation we refer to the remark after Proposition \[pro:Q2\].

[*Proof of Proposition \[pro:4\].*]{} It remains to 