is introduced as a function
such that
for any element q of Q and any element n of N.
(Of course, teachers do not use these expressions directly.)
It is the "proportional function as a tool" that is depicted in this way of introduction. Let us recognize that the mathematical meaning of "proportional function" is "homomorphism" of the quantity (i.e., function between quantities that conserve the algebraic relation). The expression that shows this meaning is
Here, in fact, the latter is deduced from the former.