Find a Context-Free Grammar for $L:=\{a^nb^mc^{n+m}\mid n,m\in\mathbb{N}\}$

I want to find a Context-Free Grammar for $ L:=\{a^nb^mc^{n+m}\mid n,m\in\mathbb{N}\}$

I’ve tried the following:

$ G=(V,\Sigma,R,S)$ with $ \Sigma=\{a,b,c,\lambda\}$ , $ V=\{S,B\}$ , $ S=S$ and $ $ R=\{S\to \lambda\mid aSc\mid B,\;B\to bBc\mid \lambda\},$ $ which would output $ L:=\{a^nb^mc^{n+m}\mid n,m\in\mathbb{N}\}$ , in my opinion. I’ve tried to test my grammar by applying the rules in different combinations and I didn’t spot any error yet.

So I’m asking myself:
Is there a way to to see if $ L(G)=L$ or do I need to assume, that I’ve done everything correctly after testing some cases?