At the beginning of the section $1.18$ of Tom Apostol's Calculus vol $1.$ (second edition), it is written that from the area properties introduced in section $1.6$, he proved that the area of the ordinate set of a nonnegative step function is equal to the integral of the function.
However, in the section $1.12$, where he is defining the integral for step functions, he just says that the definition is made such that the integral is equal to the area of the function's ordinate set, but I do not see any proof for that later.
Is there a proof maybe in an earlier version of the book, and it was removed from the second edition (which is the one I have)? In particular, I would be interested to see the proof that the step function integral satisfies the exhaustion property (i.e. Axiom $6$ of an area function).