Transposes of inverses

In the lecture of "Transposes of sums and inverses" (in Leaner Algebra / Matrix transformations), it's not obvious why "transpose of A inverse is the inverse of transpose of A".

Should it say at front that "A*(A inverse) = I & (inverse of transpose of A)*(transpose of A) = I ? Then transpose of A*(A inverse) = I = (inverse of transpose of A)*(transpose of A) can lead to  the conclusion: transpose of A inverse = inverse of transpose of A

https://youtu.be/Br_8t58mdeI?t=509