It seems like in a lot of analysis proofs of equalities via inequalities one direction is proven directly, but then the other direction will be shown by inserting, say $-f$ in for $f$ like rudin does in his proof of theorem 6.12 (a) which was typed up here :Theorem 6.12 (a) in Baby Rudin: $\int_a^b \left( f_1 + f_2 \right) d \alpha=\int_a^b f_1 d \alpha + \int_a^b f_2 d \alpha$.
I see why making the substitution above gives us the desired result, but I don't feel entirely comfortable with this method in the sense that I can't really say why it is valid. I fear I may be overthinking it. To be clear, I am not asking why rudin's proof above works, I want to understand why the general trick of substituting in a negative value in for the original is a valid way to prove the reverse inequality.