Given a coalgebra $C$ over the field $k$ and subspaces $U$ and $V$ of $C$, one can define the wedge product $U\wedge V$ of $U$ and $V$ to be $\Delta^{-1}(U\otimes C+C\otimes V)$.
Now, suppose that $C=\bigoplus_{i\in I}C_i$ is the direct sum of sub-coalgebras, and $D=\bigoplus_{i\in I}D_i$, $E=\bigoplus_{i\in I}E_i$ are sub-coalgebras of $C$, where $D_i$ and $E_i$ are sub-coalgebras of $C_i$ for each $i\in I$. We are asked to show that $D\wedge E=\bigoplus_{i\in I}D_i\wedge E_i$.
The containment $\supseteq$ is clear; however I have trouble showing the other containment. I was trying to write an element in $D\wedge E$ as a sum of elements in $\bigoplus_{i\in I}C_i$, but the problem arise when I try to show that each of them must, in fact lie in $D_i\wedge E_i$. Are there any other ways of attacking the problem?