Affine spaces/Product/Basis/Exercise

Let ${}E$ and ${}F$ be affine spaces over the field ${}K$, and let an affine basis ${}P_{1},\ldots ,P_{n}$ of ${}E$ and an affine basis ${}Q_{1},\ldots ,Q_{m}$ of ${}F$ be given. Show that

$(P_{1},Q_{1}),\,(P_{1},Q_{2}),\ldots ,(P_{1},Q_{m}),\,(P_{2},Q_{1}),\,(P_{3},Q_{1}),\ldots ,(P_{n},Q_{1})$

is an affine basis of the product space ${}E\times F$.

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