# distributive law of differential form

Suppose we have two $$p$$-form $$\omega_{1},\omega_{2}$$ and a $$q$$-form $$\lambda$$ where $$\omega_{1}=\sum_{I_{1}}b_{I_{1}}(x)dx_{I_{1}}\\\omega_{2}=\sum_{I_{2}}b_{I_{2}}(x)dx_{I_{2}}\\\lambda=\sum_{J}c_{J}(x)dx_{J}$$ where $$I_{1},I_{2},J$$ are increaseing indices in $$\{1,\dots,p\}$$ and $$\{1,\dots,q\}$$ respectively.

If $$\int_{\Phi}\omega=\int_{\Phi}\omega_{1}+\int_{\Phi}\omega_{2}$$ for every $$p$$-surface $$\Phi$$, we define $$\omega=\omega_{1}+\omega_{2}$$

Now if we define $$\omega\wedge \lambda=\sum_{I,J}b_{I}c_{J}dx_{I}\wedge dx_{J}$$

How do we use the definition above to prove $$(\omega_{1}+\omega_{2})\wedge \lambda=\omega_{1}\wedge \lambda+\omega_{1}\wedge \lambda$$

First, your notation is pretty horrible. Write \begin{align*} \omega_1 &= \sum b_I\,dx_I \quad\text{and}\\ \omega_2 &= \sum b'_I\,dx_I, \end{align*} with the same (increasing) multiindices and different coefficients.
Now show that it follows from your integration over $$p$$-surface definition that $$\omega_1+\omega_2 = \sum (b_I+b'_I)\,dx_I$$. (You will want to choose the $$p$$-surfaces to be small subsets of the coordinate $$p$$-planes. The usual way to proceed would be to assume you had some point $$p$$ and some increasing subset $$I_0$$ where the formula fails to hold.)