Relationship between Scorer Hi and Airy functions

Theorem: The following formula holds: $$\mathrm{Hi}(x)=\mathrm{Bi}(x)\displaystyle\int_{-\infty}^x \mathrm{Ai}(t)dt - \mathrm{Ai}(x)\displaystyle\int_{-\infty}^x \mathrm{Bi}(t)dt,$$ where $\mathrm{Hi}$ denotes the Scorer Gi function, $\mathrm{Ai}$ denotes the Airy Ai function, and $\mathrm{Bi}$ denotes the Airy Bi function.

Proof: █