Corollary

Let $(X_j)_{j\in J}$ and $(Y_j)_{j\in J}$ be families of topological spaces having the same index set. For each $j\in J$, let $f_j$ and $g_j$ be continuous mappings of $X_j$ into $Y_j$. If, for each $j\in J$, the mappings $f_j$ and $g_j$ are homotopic, the same is true of the product mappings $f: (x_j)\mapsto (f_j(x_j))$ and $g: (x_j)\mapsto (g_j(x_j))$ of $\prod_{j\in J}X_j$ into $\prod_{j\in J}Y_j$.

Suppose that we have $[f_j] = [g_j]$ for each $j\in J$. We have [pr$_j]\circ [f] =$ $[f_j\circ$ pr$_j]$ and [pr$_j]\circ [g] = [g_j\circ$ pr$_j]$, hence [pr$_j]\circ [f] =$ [pr$_j]\circ [g]$ by prop. 2, whence $[f] = [g]$ by prop. 3.