## 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.

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?