Given two paths $f,g: [0,1] \mapsto X $
Then their product is defined by $f\cdot g := \begin{cases} f(2t) , \space 0\leq t \leq \frac{1}{2}\\ g(2t-1) ,\space \frac{1}{2} \leq t \leq 1\\ \end{cases} $
Now it is said that $[f\cdot g] = [f]\cdot [g]$.
Where [-] is the homotopic class relative to $\{0,1\}$.
But I can't see why this is the case. Surely it could be the case where $h$ is path-homotopic to $f\cdot g$ but $h(1/2) \neq f(1)=g(0)$
Hence not every element in $[f\cdot g]$ can be represented by the product of two elements in $[f]$ and $[g]$ respectively?
Hope someone can clear this up!