The answer to this is true because that's how they worked.
What Frère Jacques has to do with Mahler's first symphony is that Frère Jacques' song was infused in the third movement of Gustav Mahler's first symphony.
Not only that, Gustav Mahler intended to change the symphony from minor key to major key as his first symphony was a funeral march.
Thus, Frere Jacques's song contains the major key, while the rest of the symphony has a minor key.
Gustav Mahler's first symphony has four-movement structures.
Hence, in this case, it is concluded that Frère Jacques' song was infused in the third movement of Gustav Mahler's first symphony.
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There's quite a lot of improvisation musicians, such as Richard Grayson, Daniel Roth, and even Beethoven composed an improvisation named, "Rage over a Lost Penny"