I've read the claim about base change for flat modules in several sources (Lang's Algebra, Hartshorne's Algebraic Geometry, A&M), but unfortunately it isn't proven anywhere. The claim is that
Let $A$ be a commutative $R$-algebra, and $F$ a flat $R$-module. Then $A\otimes_R F$ is a flat $A$-module.
The proof is supposedly immediate, but sadly not to me. Is there a nice standard proof of the claim I could read? Thank you.