Let $M$ be a manifold and $m\in M$. We call $t$ a tangent at $m$ if for every pair $(f,g)$ of smooth real functions defined in a neighborhood of $m$ and for every pair $(a,b)$ of real numbers, we have $t(af+bg)=at(f)+bt(g)$ and $t(fg)=t(f)g(m)+f(m)t(g)$. (The domain of $t$ is the the set of all smooth real maps defined in a neighborhood of $m$ and its codomain is $\mathbb R$.)
Now suppose that $M,N$ are manifolds and $p,q$ are respectively projection maps of $M\times N$ onto $M$ and $N$.
I want to show that if for all real smooth map $f$ defined in a neighborhood of $m\in M$ and for all real smooth $g$ defined in a neighborhood of $n\in N$ we have $t(f\circ p)=0, t(g\circ q)=0$ where $t$ is a tangent at $(m,n)$, then $t\equiv 0$. What is the idea here?
I already know a solution but not the solution I am looking for. This solution is easy. It follows from the answer to Maps between tangent space of product manifold and sum of tangent spaces.