I am trying to figure out how to prove the equality I circled below in red. I have figured out how to prove the text in blue but don't know how to use that to prove the equality I circled in red.

Below I will post the givens I'm trying to use and my guess of how to prove it.

P.S.I understand this only proves the chain rule in the special case where . This is from Stewart's Calculus and he does mention that this is not a full proof but I'm very curious how to prove this special case anyway.

I'm not sure if I'm using the right givens below.

In their blue form it looks like I will be able to use the transitive property of implication if I can always let the from the 1st given equal the from the second line.