Straight Line Segment is Shortest Path between Two Points/Lemma 2

Lemma to Straight Line Segment is Shortest Path between Two Points
Let:
 * $\forall x \in \closedint a b: \map f x = \map g x$

Then:
 * $\ds \int_a^b \map f x \rd x = \int_a^b \map g x \rd x$

Proof
Follows directly from the definition of the definite integral.